Introduction to Pre-Calculus

These are my notes on an Introduction to Pre-calculus.

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Functions and Notation

Intro
A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the range. consider the following set of ordered pairs.
The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first. 
\(1,2  2,4  3,6  4,8  5,10\)
The domain is (1,2,3,4,5)
The range is (2,4,6,8,10)

Note that each value in the domain is also known as an input or independent
variable. It is often labeled with a lowercase x. Each value in the range is
also known as an output value or dependent variable. It is often labeled with
a lowercase y. 

Functions
A function f is a relation that assigns a single value in the range to each
value in the domain. In other words, no x-values are repeated. For our example
that relates the first five natural numbers to numbers double their value, this
relation is a function because each element in the domain (1,2,3,4,5) is
paired with exactly one element in the range (2,4,6,8,10).

Now lets consider the set of ordered pairs that relates the terms even
and odd to the first five natural numbers.
(odd,1) (even,2)  (odd,3)  (even,4)  (odd,5)

Notice that each element in the domain (even,odd) is not paired with exactly
one element in the range, (1,2,3,4,5). For example, the term odd corresponds
to three values from the range (1,3,5) and the term even corresponds to two
values from the range (2,4). This violates the definition of a function,
so this relation is not a function. 

A function is a relation in which each possible input value leads to exactly
one output value. We say the output is a function of the input. The input
values make up the domain, and the output values make up the range. 

To decide if a relation is a function. identify the input values, identify
the output values. then: if each input value leads to only one output value,
classify the relationship as a function. If any input value leads to two or
more outputs, do not classify the relationship as a function.

Function Notation
Once that we determine that a relationship is a function, we need to
display and define the functional relationship so that we can understand and use
them, and sometimes also so that we can program them into computers. There are
various ways of representing functions. A standard function notation is one
representation that facilitates working with functions. 

To represent height is a function of age, we start by identifying the
descriptive variables "h" for height and "a" for age. The letters f,g, and h,
are often used to represent functions. 

h is f of a - We name the function f, height is a function of age.
h=f(a) - We use parentheses to indicate the function input.
f(a) - We name the function f, the expression is read as "f of a".

Remember that we can use any letter to name the function. The notation
h(a) shows us that h depends on a. The value a must be put into the function h
to get a result. The parentheses indicate that age is input into the function,
they do not indicate multiplication. 

We can also give an algebraic expression as the input to a function. For
example, f(a+b) means first a and b, and the result is the input for the
function f. The operations must be performed in this order to obtain the
correct result.

The notation y=f(x) defines a function named f. This is read as "y is a
function of x". The letter x represents the input value or independent
variable. The letter y or f(x) represents the output value or dependent
variable.

Example 1
Determine if menu price lists are functions.
Is the price a function of the item?
Is the item a function of the price?

Solution: 
Let's begin by considering the input as the items on the menu. The
output values are then the prices. Each item on the menu has only one price,
so the price is a function of the item. 

Two items on the menu have the same price. If we consider the prices to be
the input values and the items to be the output, then the same input value
could have more than one output associated with it. So, the item is not a
function of the price.

Example 2
Determine if class grade rules are functions
In a particular math class, the overall percent grade corresponds to a grade
point average. Is grade point average a function of the percent grade? Is the
percent grade a function of the grade point average?

Solution:
For any percent grade earned, there is an associated grade point average, so
the grade point average is a function of the percent grade. In other words, if
we input the percent grade, the output is a specific grade point average. 

In the grading system given, there is a range of percent grades that
correspond to the same grade point average. For example, students who
receive a grade point average of 3.0 could have a variety of percent grades
ranging from 78 all the way to 86. Thus, the percent grade is not a function
of grade point average.

Example 3
Using function notation for days in a month
Use function notation to represent a function whose input is the name of a
month and output is the number of days in that month. Assume that the domain
does not include leap years.

Solution:
The number of days in a month is a function of the name of the month, so if
we name the function f, we write days=f(month) or d-f(m). The name of the
month is the input to a rule that associates a specific number(the
output) with each input. For example, f(march)=31, because march has 31
days. The notation d-f(m) reminds us that the number of days, d(output), is
dependent on the name of the month, m(input). 

Note that the inputs to a function do not have to be numbers. Function inputs
can be the names of people, labels of geometric objects, or any other
element that determines some kind of output. However, most of the functions we
will work with in this book will have numbers as inputs and outputs. 

Example 4
Interpreting function notation
A function N=f(y) gives the number of police officers, N, in a town in year y.
What does f(2005)=300 represent?

Solution:
When we read f(2005)=300, we see that the input year is 2005. The value for
the output, the number of police officers(N), is 300. Remember N=f(y). The
statement f(2005)=300 tells us that in the year 2005 there were 300 police
officers in town.

Representing Functions Using Tables
A common method of representing functions is in the form of a table. The
table rows or columns display the corresponding input and output values. in
some cases, these values represent all we know about the relationship.; other
times, the table provides a few select examples from a more complete
relationship.

Finding Input and Output Values of a Function
When we know an input value and want to determine the corresponding output
value for a function, we evaluate the function. Evaluating will always
produce one result because each input of a function corresponds to exactly
one output value.

When we know an output value and want to determine the input values that would
produce that output value, we set the output equal to the function's formula
and solve for the input. Solving can produce more than one solution because
different input values can produce the same output value. 

When we have a function in formula form, it is usually a simple matter to
evaluate the function. For example, the function \(f(x)=5-3x^2\) can be
evaluated by squaring the input value, multiplying by 3, and then subtracting
the product from 5.

Example 6
Evaluate \(f(x)=x^2+3x-4\) for x=2

Solution:
\(f(x)=(2)^2 + 3(2) - 4\)
\(f(x)=4+6-4\)
\(f(x)=6\)

Example 7
Given the function \(h(p)=p^2+2p\) evaluate h(4)

Solution:
To evaluate h(4), we substitute the value 4 for the input variable p in the
given function.
\(h(p)=p^2+2p\)
\(h(4)=4^2+2(4)\)
\(=16+8\)
\(=24\)

Example 8
Given the function \(h(p)=p^2+2p\) solve for h(p)=3

Solution:
\(h(p)=3\)
\(p^2+2p=3\)
\(p^2+2p-3=0\)
\((p+3)(p-1)=0\)
p=-3,1

Evaluating Function Expressed in Formulas
Some functions are defined by mathematical rules or procedures expressed in
equation form. If it is possible to express the function output with a
formula involving the input quantity, then we can define a function in
algebraic form. For example, the equation \(2n+6p=12\) expresses a functional
relationship between n and p. We can rewrite it to decide if p is a function
of n. 

Given a function in equation form, write its algebraic formula:
1. Solve the equation to isolate the output variable on one side of the equal
sign, with the other side as an expression that involves only the input
variable.
2. Use all the usual algebraic methods for solving equations, such as adding
or subtracting the same quantity to or from both sides, or multiplying or
dividing both sides of the equation by the same quantity. 

Example 9
Finding the Algebraic form of a Function
Express the relationship \(2n+6p=12\) as a function \(p=f(n)\) if possible.

Solution:
To express the relationship in this form, we need to be able to write the
relationship where p is a function of n, which means writing it as
p=[expression involving n].
\(2n+6p=12\)
\(6p=12-2n\)
\(p=\frac{12-2n}{6}\)
\(p=\frac{12}{6} - \frac{2n}{6}\)
\(p=2-\frac{1}{3}n\)

It is important to note that not every relationship expressed by an equation
can also be expressed as a function with a formula.

Example 10
Expressing the Equation of a Circle as a Function
does the equation \(x^2+y^2=1\) represent a function with x as input and y as
output? If so, express the relationship as a function \(y=f(x)\).

Solution:
First we subtract \(x^2\) from both sides.
\(y^2=1-x^2\)
Now we try to solve for y in this equation
\(y=\pm \sqrt{1-x^2}\)
\(y=+ \sqrt{1-x^2} \text{ and} - \sqrt{1-x^2}\)
We get two outputs corresponding to the same input, so this relationship cannot
be represented as a single function \(y=f(x)\)

Are there relationships expressed by an equation that do represent a function
but which still cannot be represented by an algebraic formula?
Yes this can happen. For example, given the equation \(x=y+2^y\), if we want
to express y as a function of x, there is no simple algebraic formula
involving only x that equals y. However, each x does determine a unique value
for y, and there are mathematical procedures by which y can be found to any
desired accuracy. In this case, we say that the equation gives an
implicit rule for y as a function of x, even though the formula cannot be
written explicitly.

Evaluating a Function Given in Tabular Form
As we saw above, we can represent functions in tables. Conversely, we can use
information in tables to write functions, and we can evaluate functions using
the tables. 

Given a function represented by a table, identify specific output and input
values:
1. Find the given input in the row of input values
2. Identify the corresponding output value paired with that input value
3. Find the given output values in the row or column of output values, noting
every time that output value appears.
4. Identify the input value or values corresponding to the given output value.

Example 11
Evaluating and Solving a Tabular Function
1. Evaluate g(3)
2. Solve \(g(n)=6\)

Solution:
Evaluating g(3) means determining the output value of the function g for the
input value of n=3. The table output value corresponding to n=3 is
7, so g(3)=7.

Solving g(n)=6 means identifying the input values n, that produce an output
value of 6. When we input 2 into the function g, our output is 6. When we
input 4 into the function g, our output is also 6.

Finding Function Values From a Graph
Evaluating a function using a graph requires finding the corresponding value
for a given input value, only in this case, we find the output value by
looking at the graph. Solving a function equation using a graph requires
finding all instances of the given output value on the graph and observing the
corresponding input values.
1. Evaluate f(2)
2. Solve f(x)=4

Solution:
1. To evaluate f(2), locate the point on the curve where x=2, then read the
y-coordinate of that point. The point has coordinates of (2,1) so f(2)=1.
2. To solve f(x)=4, we find the output value 4 on the vertical axis. moving
horizontally along the line y=4, we locate two points of the curve with output
value 4:(-1,4) and (3,4). These points represent the two solutions to f(x)=4:
-1 or 3. This means f(-1)=4 and f(3)=4, or when the input is -1 or 3, the
output is 4.

Determining Whether a Function is One-to-One
Some functions have a given output value that corresponds to two or
more input values. For example, in the stock chart shown, the stock price
was $1000 on five different dates, meaning that there were five different input
values that all resulted in the same output value of $1000.

However, some functions have only one input value for each output value, as
well as having only one output for each input. We call these functions
one-to-one functions. As an example, consider a school that uses only letter
grades and decimal equivalents.

This grading system represents a one to one function, because each letter input
yields one particular grade point average output and each grade point average
corresponds to one input letter.

To visualize this concept, let's look again at the two simple functions
sketched. The function in part 1 shows a relationship that is not a one to one
function because inputs q and r both give output n. The function in part 2
shows a relationship that is a one to one function because each input is
associated with a single output.

A one to one function is a function in which each output value corresponds
to exactly one input value.

Example 13
Determining Whether a Relationship is a One to One function
Is the area of a circle a function of its radius? if yes, is the function one
to one?

Solution:
A circle of radius r has a unique area measure given by \(A=\pi r^2\) for any
input r, there is only one output. The area is a function of radius r.

If the function is one to one, the output value, the area, must correspond to
a unique input value, the radius. Any area measure A is given by the formula
\(A=\pi r^2\). Because areas and radii are positive numbers, there is exactly
one solution: \(\sqrt{A}{\pi}\). So the area of a circle is a one to one
function of the circle's radius. 

Using the Vertical Line Test
As we have seen in some examples above, we can represent a function
using a graph. Graphs display a great many input-output pairs in a small
space. The visual information they provide often makes relationships easier to
understand. by convention, graphs are typically constructed with the
input values along the horizontal axis and the output values along the vertical
axis.

The most common graphs name the input value y, and we say y is a function of x,
or y=f(x) when the function is named f. The graph of the function is the set
of all points (x,y) in the plane that satisfies the equation y=f(x). If the
function is defined for only a few input values, then the graph of the function
is only a few points, where the x-coordinate of each point is an input value
and the y-coordinate of each point is the corresponding output
value. for example, the black dots on the graph tell us that f(0)=2 and
f(6)=1. However, the set of all points (x,y) satisfying y=f(x) is a
curve. The curve includes (0,2) and (6,1) because the curve passes through
those points.

The vertical line test can be used to determine whether a graph
represents a function. If we can draw any vertical line that intersects a
graph more than once, then the graph does not define a function because a
function has only one output value for each input value.

Given a graph, use the vertical line test to determine if the graph represents a
function.
1. Inspect the graph to see if any vertical line drawn would intersect the curve
more than once.
2. If there is any such line, determine that the graph does not represent a
function.

Example 14
Applying the vertical line test
Which of the graphs represent a function y=f(x)?

Solution:
If any vertical line intersects a graph more than once, the relation
represented by the graph is not a function. Notice that any vertical line
would pass through only one point of the two graphs shown in parts a and b.
From this we can conclude that these two graphs represent functions. The
third graph does not represent a function because, at most x-values, a vertical
line would intersect the graph at more than one point.

Using the Horizontal line test
Once we have determined that a graph defines a function, an easy way to
determine if it is a one to one function is to use the horizontal line test. Draw
horizontal lines through the graph. If any horizontal line intersects the graph
more than once, then the graph does not represent a one to one function.

Given a graph of a function, use the horizontal line test to determine if the
graph represents a one to one function.
1. Inspect the graph to see if any horizontal line drawn would intersect the
curve more than once.
2. If there is any such line, determine that the function is not one to
one.

Example 15
Applying the Horizontal Line Test
Consider the functions shown. Are either of the functions one to one?

Solution:
1. The function in part 1 is not one to one. The horizontal line shown
intersects the graph of the function at two points.
2. The function in part 2 is one to one. Any horizontal line will intersect a
diagonal line at most once.

Identifying Basic Toolkit Functions
In this section, we will be exploring functions-the shapes of their graphs,
their unique characteristics, their algebraic formulas, and how to
solve problems with them. When learning to read, we start with the alphabet.
When learning to do arithmetic, we start with numbers. When working with
functions, it is similarly helpful to have a base set of building block
elements. We call these our toolkit functions, which form a set of basic named
functions for which we known the graph, formula, and special properties. Some
of these functions are programmed to individual buttons on many
calculators. For these definitions, we will use the x as the input variable and
y=f(x) as the output variable.

We will see these toolkit functions, combinations of toolkit functions, their
graphs, and their transformations frequently throughout this book. It
will be very helpful if we can recognize these toolkit functions and their
features quickly by name, formula, graph, and basic table properties. The
graphs and sample table values are included with each function shown. 

Domain and Range

Finding The Domain Of A Function
In determining domains and ranges, we need to consider what is physically
possible in real world examples. We also need to consider what is
mathematically permitted. For example, we cannot include any input value
that leads us to take an even root of a negative number if the domain and
range consist of real numbers. Or in a function expressed as a formula,
we cannot include any input value in the domain that would lead us to divide
by 0.

We can visualize the domain as a holding area that contains raw materials
for a function machine and the range as another holding area for the
machine's products.

We can write the domain and range in interval notation, which uses
values within brackets to describe a set of numbers. In interval notation, we
use a square bracket when the set includes the endpoint and a
parenthesis to indicate the endpoint is either not included or the interval
is unbounded. For example, if a person has $100 to spend, they would need to
express the interval that is more than - and less than or equal to 100 and write
(0,100].

Let us turn our attention to finding the domain of a function whose equation
is provided. Oftentimes, finding the domain of such functions involves
remembering three different forms. First, if the function has no
denominator or an odd root, consider whether the domain could be all real
numbers. Second, if there is a denominator in the function's equation,
exclude values in the domain that force the denominator to be zero.
Third, if there is an even root, consider excluding values that would make
the radicand negative.

The smallest term from the interval is written first.
The largest term in the interval is written second, following a comma.
Parentheses are used to signify that an endpoint is not included, called
exclusive.
Brackets are used to indicate that an endpoint is included, called
inclusive.

Example 1
Find the domain of the following function:
{(2,10), (3,10), (4,20), (5,30),(6,40)}

Solution:
First, identify the input values. The input value is the first coordinate in
an ordered pair. There are no restrictions, as the ordered pairs are simply
listed. The domain is the set of the first coordinates of the ordered pairs.
{2,3,4,5,6}

Example 2
Find the domain of the function \(f(x)=x^2-1\)

Solution:
The input value, shown by the variable x in the equation, is squared and
then the result is lowered by 1. Any real number may be squared and then be
lowered by 1, so there are no restrictions on the domain of this function.
The domain is the set of real numbers.
In interval form, the domain of f is: (-inf,inf)

Example 3
Find the domain of the function: 
\(f(x)=\frac{x+1}{2-x}\)

Solution:
When there is a denominator, we want to include only values that do not force
the denominator to be zero. So, we will set the denominator equal to 0
and solve for x.
\(2-x=0\)
\(-x=-2\)
\(x=2\)
Now, we will exclude 2 from the domain. The answers are all real numbers
where x<2 or x>2. We can use a symbol known as the union to combine the two
sets. In interval notation, we write the solution:
\((-\infty,2) \cup (2,\infty)\)

Example 4
Find the domain of the function \(f(x)=\sqrt{7-x}\).

Solution:
When there is an even root in the formula, we exclude any real numbers that
result in a negative number in the radicand.
Set the radicand greater than or equal to zero and solve for x
\(7-x \geq 0\)
\(-x \geq -7\)
\(x \leq 7\)
Now, we will exclude any number greater than 7 from the domain. The answers
are all real numbers less than or equal to 7.
\((-\infty,7)\)

Can there be functions in which the domain and range do not intersect at all?
Yes, for example, the function \(f(x)=-\frac{1}{\sqrt{x}}\) has
the set of all positive real numbers as its domain but the set of all negative
real numbers as its range. As a more extreme example, a function's inputs and
outputs can be completely different categories, in such cases the domain and
range have no elements in common.

Using Notations to Specify Domain and Range
In the previous examples, we used inequalities and lists to describe the
domain of functions. We can also use inequalities, or other statements that
might define sets of values or data, to describe the behavior of the variable
in set-builder notation. For example, \({x|10 \leq x < 30}\) describes the
behavior of x in set-builder notation. The braces "{}" are read as "the set of",
and the vertical bar "|" is read as "such that" so we would read it as "the set
of x-values such that 10 is less than or equal to x, and x is less than 30."

Here are some examples of different notation:
\(5<h\leq10\) : \({h|5<h\leq10}\) : \(5,10]\)
\(5\leq h<10\) : \({h|5\leq h<10}\) : \([5,10)\)
\(5<h<10\)    : \({h|5<h<10}\)    : \((5,10)\)
\(h<10\)      : \({h|h<10}\)      : \((-\infty,10)\)
all numbers   : \({R}\)             : \((-\infty,\infty)\)

To combine two intervals using inequality notation or set-builder notation, we
use the word "or". As we saw in earlier examples, we use the union symbol,
\(\cup\), to combine two unconnected intervals. For example, the union of
sets {2,3,5} and {4,6} is the set {2,3,4,5,6}. It is the set of all elements
that belong to one or the other(or both) of the original two sets. For
sets with a finite number of elements like these, the elements do not have to
be listed in ascending order of numerical value. If the original two sets
have some elements in common, those elements should be listed only once in the
union set. For sets of real numbers on intervals, another example of a
union is: \({x| |x| \geq 3} = (-\infty,-3) \cup [3,\infty)\)

Set-Builder Notation
This is a method of specifying a set of elements that satisfy a certain
condition. It takes the form \({x| ... x}\) which reads as, "the set of all
x such that the statement about x is true."
\({x|4<x\leq12}\)

Interval Notation
This is a way of describing sets that include all real numbers between a lower
limit that may or may not be included and an upper limit that may or may
not be included. The endpoint values are listed between brackets or
parentheses. A square bracket indicates inclusion in the set, and a
parenthesis indicates exclusion from the set. 
\((4,12]\)

Given a line graph, describe the set of values using interval notation.
1. Identify the intervals to be included in the set by determining where the
heavy line overlays the real line.
2. At the left end of each interval, use [ with each end value to be included
in the set (solid dot) or a ( for each excluded end value (open dot).
3. At the right end of each interval, use ] with each end value to be included
in the set (filled dot) or ) for each excluded end value (open dot).
4. Use the union symbol \(\cup\) to combine all intervals into one set.

Example 5
Describe the intervals of values shown using inequality notation,
set-builder notation, and interval notation. 

Solution:
To describe the values, x, included in the intervals shown, we would say "x is
a real number greater than or equal to 1 and less than or equal to 3, or a real
number greater than 5."
Inequality:  \(1 \leq x \leq 3 \text{ or} x > 5\)
Set-Builder: \({x| 1 \leq x \leq 3 \text{ or} x > 5}\)
interval:    \([1,3] \cup (5,\infty)\)
Remember that, when writing or reading interval notation, using a square
bracket means the boundary is included in the set. Using a parenthesis means
the boundary is not included in the set.

Finding Domain and Range From Graphs
Another way to identify the domain and range of functions is by using graphs.
Because the domain refers to the set of possible input values, the domain of a
graph consists of all input values shown on the x-axis. The range is the
set of all possible values, which are shown on the y-axis. Keep in mind that
if the graph continues beyond the portion of the graph we can see, the domain
and range may be greater than the visible values. We can observe that the
graph extends horizontally from -5 to the right without bound, so the
domain is \([-5,\infty)\). The vertical extent of the graph is all range values
5 and below, so the range is \((-\infty, 5]. Note that the domain and range are
always written from smaller to larger values, or from left to right for domain,
and from the bottom of the graph to the top of the graph for range.

Example 6
Find the domain and range of the function f whose graph is shown.

Solution:
We can observe that the horizontal extent of the graph is -3 to 1, so the domain
of f is (-3,1].
The vertical extent of the graph is 0 to -4, so the range is [-4,0].

Example 7
Find the domain and range of the function f whose graph is shown.

Solution:
The input quantity along the horizontal axis is years, which we represent with
the variable T for time. The output quantity is thousands of barrels a
per day, which we represent with the variable b for barrels. The graph may
continue to the left and right beyond what is viewed, but based on the portion
of the graph that is visible, we can determine the domain as \(1973 \leq t
\leq 2008\) and the range as approximately \(180 \leq b \leq 2010\).

In interval notation, the domain is [1973,2008] and the range is about
[180,2010]. for the domain and the range, we approximate the smallest
and largest values since they do not fall exactly on the grid lines. 

Can a function's domain and range be the same?
Yes. For example, the domain and range of the cube root function are both
the set of all real numbers. 

Finding Domains and Ranges of the Toolkit Functions
We will now return to our set of toolkit functions to determine the domain and
range of each.

Constant Function 
\(f(x)=c\)  
D=\((-\infty,\infty)\) : R=\([c,c]\)
For the constant function f(x)=c, the domain consists of all real numbers.
There are no restrictions on the input. The only output value is the
constant c, so the range is the set {c} that contains this single element. In
interval notation, this is written as [c,c], the interval that both begins
and ends with c.

Identity Function
\(f(x)=x\)
D=\((-\infty,\infty)\) : R=\((-\infty,\infty)\)
For the identity function f(x)=x, there is no restriction on x. Both the
domain and and range are the set of all real numbers.

Absolute Function
\(f(x)=|x|\)
D=\((-\infty,\infty)\) : R=\([0,\infty)\)
For the absolute value function, \(f(x)=|x|\), there is no restriction on x.
However, because absolute value is defined as a distance from 0, the
output can only be greater than or equal to 0.

Quadratic Function
\(f(x)=x^2\)
D=\((-\infty,\infty)\) : R=\([0,\infty)\)
For the quadratic function \(f(x)=x^2\), the domain is all real numbers since
the horizontal extent of the graph is the whole real number line. Because the
graph does not include any negative value for the range, the range is only
nonnegative real numbers. 

Cubic Function
\(f(x)=x^3\)
D=\((-\infty,\infty)\) : R=\((-\infty,\infty)\)
For the cubic function \(f(x)=x^3\), the domain is all real numbers because the
horizontal extent of the graph is the whole real number line. The same
applies to the vertical extent of the graph, so the domain and range includes
all real numbers.

Reciprocal Function
\(f(x)=\frac{1}{x}\)
D=\((-\infty,0) \cup (0,\infty)\) : R=\((-\infty,0) \cup (0,\infty)\)
We cannot divide by 0, so we must exclude 0 from the domain. further, 1
divided by any value can never be 0, so the range  also will not include 0.

Reciprocal Squared Function
\(f(x)=\frac{1}{x^2}\)
D=\((-\infty,0) \cup (0,\infty)\) : R=\((0,\infty)\)
We cannot divide by 0, so we must exclude 0 from the domain. There is also no
x that can give an output of 0, so 0 is excluded from the range as well. Note
that the output of this function is always positive due to the square in the
denominator, so the range includes only positive numbers. 

Square Root Function
\(f(x)=\sqrt{x}\)
D=\([0,\infty)\) : R=\([0,\infty)\)
We cannot take the  square root of a negative real number, so the domain
must be 0 or greater. The range also excludes negative numbers because the
square root of a positive number x is defined to be positive, even though the
square of the negative number \(-\sqrt{x}\) also gives us x.

Cube Root Function
\(f(x)=\sqrt[3]{x}\)
D=\((-\infty,\infty)\) : R=\((-\infty,\infty)\)
The domain and range include all real numbers. Note that there is no problem
taking a cube root, or any odd-integer root, of a negative number, and the
resulting output is negative(it is an odd function).

Given the formula for a function, determine the domain and range
1. Exclude from the domain any input values that result in division by 0.
2. Exclude from the domain any input values that have nonreal or undefined
number outputs.
3. Use the valid input values to determine the range of the output values.
4. Look at the function graph and table values to confirm the actual
function behavior.

Example 8
Find the domain and range of \(f(x)=2x^3-x\)

Solution:
There are no restrictions on the domain, as any real number may be cubed and
then subtracted from the result. The domain is \((-\infty,\infty)\) and the
range is also \((-\infty,\infty)\).

Example 9
Find the domain and range of \(f(x)=\frac{2}{x+1}\).

Solution:
We cannot evaluate the function at -1 because division by zero is undefined. The
domain is \((-\infty,-1)\cup(-1,\infty)\). Because the function is never 0, we
exclude 0 from the range. The range is \((-\infty,0)\cup(0,\infty)\).

Example 10
Find the domain and range of \(f(x)=2\sqrt{x+4}\).

We cannot take the square root of a negative number, so the value inside the
radical must be nonnegative.
\(x+4 \geq 0 \text{ when } x \geq -4\)
The domain of \(f(x) \text{ is } [-4,\infty)\).
We then find the range. We know that f(-4)=0, and the function value increases
as x increases without any upper limit. We conclude that the range of f is
\([0,\infty)\). 

Graphing Piecewise-Defined Functions
Sometimes, we come across a function that requires more than one formula in
order to obtain the given output. For example, in the toolkit functions, we
introduced the absolute value function \(f(x)=|x|\). With a domain of all real
numbers and a range of values greater than or equal to 0, absolute value can be
defined as the magnitude, or modulus, of a real number value regardless of sign.
It is the distance from 0 on the number line. All of these definitions require
the output to be greater than or equal to 0.
If we input 0, or a positive value, the output is the same as the input.
\(f(x)=x \text{ if } x \geq 0\)
If we input a negative value, the output is the opposite of the input.
\(f(x)=-x \text{ if } x < 0\)

Because this requires two different processes or pieces, the absolute
value function is an example of a piecewise function. A piecewise function is a
function in which more than one formula is used to define the output over
different pieces of the domain. 

We use piecewise functions to describe situations in which a rule or
relationship changes as the input value crosses certain boundaries. For
example, we often encounter situations in business for which the cost per piece
of a certain item is discounted once the number ordered exceeds a certain
value. Tax brackets are another real-world example of piecewise functions. For
example, consider a simple tax system in which incomes up to $10,000 are taxed
at 10%, and any additional income is taxed at 20%. The tax on a total income S
would be \(0.1S \text{ if } S \leq 10,000 \text{ and } 1000+0.2(S-10000) \text{
if } S > 1000\).

Piecewise Function
A piecewise function is a function in which more than one formula is used to
define the output. Each formula has its own domain, and the domain of the
function is the union of all these smaller domains. 

Given a piecewise function, write the formula and identify the domain for
each interval.
1. Identify the intervals for which different rules apply.
2. Determine formulas that describe how to calculate an output from an input
in each interval.
3. Use braces and if-statements to write the function.

Example 11
Writing a Piecewise Function
A museum charges $5 per person for a guided tour with a group of 1-9 people or
a fixed $50 fee for a group of 10 or more people. Write a function relating
the number of people, n, to the cost, c.

Solution:
Two different formulas will be needed. For n-values under 10, c=5n. For
values of n that are 10 or greater, c=50.
\[c(n)= \begin{cases} 5n \text{ if } 0 < n < 10 & \quad \\ 50 \text{ if } n
\geq 10 & \quad \end{cases}\]

Example 12
A cell phone company uses the function below to determine the cost, c, in
dollars for g gigabytes of data transfer.
\[c(g)= \begin{cases} 25 \text{ if } 0 < g < 2 & \quad \\ 25 + 10(g-2) \text{ if
} g \geq 2 & \quad \end{cases}\]
Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes
of data.

Solution:
To find the cost of using 1.5 gigabytes of data, c(1.5), we first look to see
which part of the domain our input falls in. Because 1.5 is less than 2, we use
the first formula.
\(c(1.5)=$25\)
To find the cost of using 4 gigabytes of data, c(4), we see that our input of
4 is greater than 2, so we use the second formula.
\(c(4)=25+10(4-2)=$45\)

Given a piecewise function, sketch a graph
1. Indicate on the x-axis the boundaries defined by the intervals on each
piece of the domain.
2. For each piece of the domain, graph on that interval using the
corresponding equation pertaining to that piece. Do not graph two functions
over one interval because it would violate the criteria of a function.

Example 13
Sketch a graph of the function:
\[f(x)=\begin{cases} x^2 \text{ if } x \leq 1 & \quad \\ 3 \text{ if } 1 < x
\leq 2 & \quad \\ x \text{ if } x > 2 & \quad \end{cases}\]

Solution:
Each of the component functions is from our library of toolkit functions, so we
know their shapes. We can imagine graphing each function and then
limiting the graph to the indicated domain. At the endpoints of the domain,
we draw open circles to indicate where the endpoint is not included because of a
less-than or greater-than inequality. We draw a closed circle where the endpoint
is included because of a less-than-or-equal-to or  greater-than-or-equal-to
inequality.

Can more than one formula from a piecewise function be applied to a value in the
domain?
No. Each value corresponds to one equation in a piecewise formula.

Rates of Change

Finding the Average Rate of Change of a Function
The price change per year is a rate of change because it describes how an
output quantity changes relative to the change in the input quantity. We can
see that the price of gasoline in the table did not change by the same amount
each year, so the rate of change was not constant. If we use only the beginning
and ending data, we would be finding the average rate of change over the
specified period of time. To find the average rate of change, we divide the
change in the output value by the change in the input value.

Average rate of change=change in output/change in input
\(=\frac{\Delta y}{\Delta x}\)
\(\frac{y_2 - y_1}{x_2 - x_1}\)
\(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\)

The Greek letter \(\Delta\) (delta) signifies the change in a quantity. We
read the ratio as "delta-y over delta-x" or "the change in y divided by
the change in x". Occasionally we write \(\Delta f\) instead of \(\Delta y\),
which still represents the change in the function's output value resulting
from a change to its input value. It does not mean we are changing the function
into some other function. 

In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7
years, the average rate of change was:
\(\frac{\Delta y}{\Delta x} = \frac{$1.37}{7years} = 0.196 \text{ dollars per
year }\)
On average, the price of gas increased by about 19.6 cents per years.

Rate Of Change
A rate of change describes how an output quantity changes relative to the
change in the input quantity. The units on a rate of change are "output units
per input units".
The average rate of change between two input values is the total change of the
function values (output values) divided by the change in the input values.
\(\frac{\Delta y}{\Delta x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\)

Given the value of a function at different points, calculate the average
rate of change of a function for the interval between two values x1 and x2.
1. Calculate the difference of \(y_2-y_1=\Delta y\)
2. Calculate the difference of \(x_2-x_1=\Delta x\)
3. Find the ratio \(\frac{\Delta y}{\Delta x}\)

Example 1
Find the average rate of change of the price of gasoline between 2007 and
2009.

Solution:
In 2007, the price of gasoline was $2.84. In 2009, the cost was $2.41. The
average rate of change is:
\(\frac{\Delta y}{\Delta x} = \frac{2.41-2.84}{2009-2007} = \frac{-0.43}{2} =
-$0.22\) per year
Note that a decrease is expressed by a negative change or negative increase. A
rate of change is negative when the output decreases as the input increases or
when the output increases as the input decreases.

Example 2
Computing average rate of change from a graph
Given the function g(t), find the average rate of change on the interval
[-1,2].

Solution:
At t=-1, the graph shows g(-1) =4. At t=2, the graph shows g(2)=1.
The horizontal change \(\Delta t= 3\) is shown by the red arrow, and the
vertical change \(\Delta g(t)=-3\) is shown by the blue arrow. The output
changes by -3 while the input changes by 3, giving an average rate of change
of:
\(\frac{1-4}{2-(-1)} = \frac{-3}{3} = -1 \)
Note that the order we choose is very important. If, for example, we use
\(\frac{y_2-y_1}{x_1-x_2}\), we will not get the correct answer. Decide
which point will be 1 and which point will be 2, and keep the coordinates
fixed as \((x_1,y_1)\) and \(x_2,y_2)\).

Example 3
Computing Average Rate of Change from a Table
After picking up a friend who lives 10 miles away, anna records her distance
from home over time. The values are shown in the table. Find her average
speed over the first 6 hours.
t(hours)    =0  1  2  3   4   5   6   7
D(t)(miles) =10 55 90 153 214 240 292 300

Solution:
Here, the average speed is the average rate of change. She traveled 282 miles in
6 hours, for an average speed of:
\(\frac{292-10}{6-0} = \frac{282}{6} = 47\)
Because the speed is not constant, the average speed depends on the interval
chosen. For the interval[2,3], the average speed is 63 miles per hour.

Example 4
Computing Average Rate of Change for a Function Expressed as a Formula
Compute the average rate of change of \(f(x)=x^2-\frac{1}{x}\) on the
interval [2,4].

Solution:
We can start by computing the function values at each endpoint of the interval.
\(f(2)=2^2-1/2 = 4-1/2 = 7/2\)
\(f(4)=4^2-1/4 = 16-1/4 = 63/4\)
Now we compute the average rate of change:
\(\frac{f(4)-f(2)}{4-2} = 49/8\)

Example 5
Finding the Average Rate of Change of a Force
The electrostatic force F, measured in newtons, between two charged
particles can be related to the distance between the particles d, in
centimeters, by the formula \(F(d)=\frac{2}{d^2}\). Find the average rate of
change of force if the distance between the particles is increased from 2 cm to
6 cm.

Solution:
We are computing the average rate of change of \(F(d)=\frac{2}{d^2}\) on
the interval [2,6].
Average rate of change:
\(\frac{F(6)-F(2)}{6-2} = \frac{(2/36)-(2/4)}{6-2} = \frac{-(16/36)}{4} =
-(1/9)\)
The average rate of change is -(1/9) newton per centimeter.

Example 6
Finding an Average Rate of Change as an Expression
Find the average rate of change of \(g(t)=t^2+3t+1\) on the interval [0,a].
The answer will be an expression involving a.

Solution:
We use the average rate of change formula.
\(\frac{g(a)-g(0)}{a-0}\)
\(\frac{(a^2+3a+1)-(0^2+3(0)+1)}{a-0}\)
\(\frac{a^2+3a+1-1}{a} = \frac{a(a+3)}{a}=a+3\)
This result tells us the average rate of change in terms of "a" between t=0
and any other point t=a. For example, on the interval [0,5], the average rate
of change would be 5+3=8.

Using a Graph to Determine Where a Function is Increasing, Decreasing,
or Constant
As part of exploring how functions change, we can identify intervals over
which the function is changing in specific ways. We say that a function is
increasing on an interval if the function values increase as the input
increases within that interval. Similarly, a function is decreasing on an
interval if the function values decrease as the input values increase over that
interval. The average rate of change of an increasing function is positive, and
the average rate of change of a decreasing function is negative. 

While some functions are increasing or decreasing over their entire domain, many
others are not. A value of the input where a function changes from increasing
to decreasing (as we go from left to right, that is, as the input variable
increases) is the location of a local maximum. The function value at that
point is the local maximum. If a function has more than one, we say it has
local maxima. Similarly, a value of the input where a function changes from
decreasing to increasing as the input variable increases is the location
of a local minimum. The function value at that point is the local minimum. The
plural form is local minima. Together, local maxima and local minima are
called local extrema, or local extreme values, of the function. The singular
form is extrenum. Often, the term local is replaced by the term relative.

Clearly, a function is neither increasing nor decreasing on an interval
where it is constant. A function is also neither increasing nor decreasing
at extrema. Note that we have to speak of local extrema, because any
given local extrenum as defined here is not necessarily the highest
maximum or lowest minimum in the function's entire domain.

For the function whose graph is shown, the local maximum is 16, and it
occurs at x=-2. The local minimum is -16 and it occurs at x=2.

To locate the local maxima and minima from a graph, we need to observe the
graph to determine where the graph attains its highest and lowest points,
respectively, within an open interval. Like the summit of a roller coaster, the
graph of a function is higher at a local maximum than at nearby points on both
sides. The graph will also be lower at a local minimum than at neighboring
points. These observations lead us to a formal definition of local
extrema.

Local Minima and Local Maxima
A function f is an increasing function on an open interval if f(b)>f(a) for
every two input values a and b in the interval where b>a.
A function f is a descending function on an open interval if f(b)<f(a) for
every two input values a and b in the interval where b>a.
A function f has a local maximum at a point b in an open interval (a,c) if
\(f(b) \geq f(x)\) for every point x (x does not equal b) in the interval. F has
a local minimum at a point b in (a,c) if \(f(b) \leq f(x)\) for every point x
(x does not equal b) in the interval.

Example 7
Finding increasing and Decreasing intervals on a Graph
Given the function p(t), identify the intervals on which the function
appears to be increasing.

Solution:
We see that the function is not constant on any interval. The function is
increasing where it slants upward as we move to the right and decreasing
where it slants downward as we move to the right. The function appears to be
increasing from t=1 to t=3 and from t=4 on.
In interval notation, we would say the function appears to be increasing
on the interval (1,3) and the interval (4,\infty).

Notice in this example that we used open intervals (intervals that do not
include the endpoints), because the function is neither increasing nor
decreasing at t=1, t=3, and t=4. These points are the local extrema(two minima
and a maximum).

Example 8
Finding Local Extrema from a Graph
Graph the function \(f(x)=\frac{2}{x} + \frac{x}{3}\). Then use the graph to
estimate the local extrema of the function and to determine the intervals
on which the function is increasing.

Using technology, it appears there is a low point, or local minimum, between
x=2 and x=3, and a mirror-image high point, or local maximum, somewhere
between x=-3 and x=-2.

Most graphing calculators and graphing utilities can estimate the location of
maxima and minima. 

Example 9
Finding Local Maxima and Minima from a Graph
For the function f whose graph is shown, find all local maxima nd minima.

Solution:
Observe the graph of f. The graph attains a local maximum at x=1 because it
is the highest point in an open interval around x=1. The local maximum is the
y-coordinate at x=1, which is 2.
The graph attains a local minimum at x=-1 because it is the lowest point in an
open interval around x=-1. The local minimum is the y-coordinate x=-1, which is
-2.

Analyzing the Toolkit Functions for Increasing or Decreasing Intervals
Constant Function 
\(f(x)=c\)
Neither increasing nor decreasing

Identity Function 
\(f(x)=x\)
Increasing

Quadratic Function
\(f(x)=x^2\)
Increasing on (0,inf)
Decreasing on (-inf,0)
Minimum at x=0

Cubic Function
\(f(x)=x^3\)
Increasing

Reciprocal Function
\(f(x)=\frac{1}{x}\)
Decreasing (-inf,0) union (0,inf)

Reciprocal Squared Function
\(f(x)=\frac{1}{x^2}\)
Increasing on (-inf,0)
Decreasing on (0,inf)

Cube Root Function
\(f(x)=\sqrt[3]{x}\)
Increasing

Square Root Function
\(f(x)=\sqrt{x}\)
Increasing on (0,inf)

Absolute Value Function
\(f(x)=|x|\)
Increasing on (0,inf)
Decreasing on (-inf,0)

Use a Graph to Locate the Absolute Maximum and Absolute Minimum
There is a difference between locating the highest and lowest points on a graph
in a region around an open interval(locally) and locating the highest and
lowest points on the graph for the entire domain. The y-coordinates (output) at
the highest and lowest points are called the absolute maximum and absolute
minimum, respectively.

To locate absolute maxima and minima from a graph, we need to observe the graph
to determine where the graph attains its highest and lowest points on the
domain of the function. Not every function has an absolute maximum or
minimum value. The function \(f(x)=x^3\) is one such function. 

Absolute Maxima and Minima
The absolute maximum of f at x=c is f(c) where \(f(c) \geq f(x)\) for all x in
the domain of f.
The absolute minimum of f at x=d is f(d) where \(f(d) \leq f(x)\) for all x in
the domain of f.

Example 10
Finding Absolute Maxima and Minima from a Graph
For the function f, find all absolute maxima and minima.

Solution:
Observe the graph of f. The graph attains an absolute maximum in two locations,
x=-2 and x=2, because at these locations, the graph attains its highest point
on the domain of the function. The absolute maximum is the y-coordinate at x=-2
and x=2, which is 16.

The graph attains an absolute minimum at x=3, because it is the lowest point on
the domain of the function's graph. The absolute minimum is the
y-coordinate at x=3, which is -10.

Composition of Functions

Combining Functions Using Algebraic Operations
Function composition is only one way to combine existing functions.
Another way is to carry out the usual algebraic operations on functions,
such as addition, subtraction, multiplication, and division. We do all this
by performing the operations with the function outputs, defining the
result as the output of our new function.

Suppose we need to add two columns of numbers that represent a husband and
wife's separate annual incomes over a period of years, with the result being
their total household income. We want to do this for every year, adding only
that year's incomes and then collecting all the data in a new column. If w(y)
is the wife's income and h(y) is the husband's income in year y, and we want T
to represent the total income, then we can define a new function.
\(T(y)=h(y)+w(y)\)
If this holds true for every year, then we can focus on the relation between
the functions without reference to a year and write:
\(T=h+w\)
Just as for this sum of two functions, we can define difference, product, and
ratio functions for any pair of functions that have the same kind of inputs
(not necessarily numbers) and also the same kinds of outputs (which do have to
be numbers so that the usual operations of algebra can apply to them, and which
also must have the same units or no units when we add and subtract). In this
way, we can think of adding, subtracting, multiplying, and dividing functions.

For two functions f(x) and g(x) with real number outputs, we define new
functions f+g, f-g, fg, and f/g by the relations:
\((f+g)(x)=f(x)+g(x)\)
\((f-g)(x)=f(x)-g(x)\)
\((fg)(x)=f(x)g(x)\)
\((f/g)(x)=f(x)/g(x)\)

Example 1
Performing Algebraic Operations on Functions
Find and simplify the functions \((g-f)(x)\) and \((g/f)(x)\), given
\(f(x)=x-1\) and
\(g(x)=x^2-1\). Are they the same function?

Solution:
begin by writing the general form, and then substitute the given functions.
\((g-f)(x) = g(x)-f(x)\)
\((g-f)(x) = x^2-1-(x-1)\)
\((g-f)(x) = x^2-x\)
\((g-f)(x) = x(x-1)\)

\((g/f)(x) = \frac{g(x)}{f(x)}\)
\((g/f)(x) = \frac{x^2-1}{x-1}\)
\((g/f)(x) = \frac{(x+1)(x-1)}{x-1}\)
\((g/f)(x) = x+1\)

No, the functions are not the same.
Note, for (g/f)(x), the condition \(x \not = 1\) is necessary because when
x=1, the denominator is equal to 0, which makes the function undefined.

Create a Function by Composition of Functions
Performing algebraic operations on functions combines them into a new function,
but we can also create functions by composing functions. When we wanted
to compute a heating cost from a day of the year, we created a new function
that takes a day as input and yields a cost as output. The process of combining
functions so that the output of one function becomes the input of another is
known as the composition of functions. The resulting function is known as a
composition function. We represent this combination by the following notation:

\(f \circ g)(x)=f(g(x))\)

We read the left-hand side as "f composed with g at x", and the right=hand side
as "f of g of x". The two sides of the equation have the same mathematical
meaning and are equal. The open circle is called the composition operator. We
use this operator mainly when we wish to emphasize the relationship between the
functions themselves without referring to any particular input value.
Composition is a binary operation that takes two functions and forms a new
function, much as addition or multiplication takes two numbers and gives a new
number. However, it is important not to confuse function composition with
multiplication because, as we learned above, in most cases
\(f(g(x))\not=f(x)g(x)\).

It is also important to understand the order of operations in evaluating a
composite function. We follow the usual convention with parentheses by starting
with the innermost parentheses first, and then working to the outside. In the
equation above, the function g takes the input x first and yields an output
g(x). Then the function f takes g(x) as an input and yields an output f(g(x)).

g(x), the output of g is the input of f

In general, \(f \circ g\) and \(g \circ f\) are different functions. in other
words, in many cases, \(f(g(x)) \not= g(f(x))\) for all x. We will also see that
sometimes two functions can be composed only in one specific order.

For example, if \(f(x)=x^2\) and \(g(x)=x+2\), then:
\(f(g(x))=f(x+2) = (x+2)^2 = x^2+4x+4\)
but:
\(g(f(x))=g(x^2) = x^2+2\)

These expressions are not equal for all values of x, so the two functions are
not equal. It is irrelevant that the expressions happen to be equal for the
single input value \(x=1/2\).

Note that the range of the inside function (first function to be evaluated)
needs to be within the domain of the outside function. Less formally, the
composition has to make sense in terms of inputs and outputs.

Composition of Functions
When the output of one function is used as the input of another function, we
call the entire operation a composition of functions. For any input x and
functions f and g, this action defines a composite function, which we write as
\(f \circ g\) such that \((f \circ g)(x)=f(g(x))\).

The domain of the composite function \(f \circ g\) is all x such that x is in
the domain of g and g(x) is in the domain of f. It is important to realize that
the product of functions fg is not the same as the function composition f(g(x)),
because , in general, \(f(x)g(x)\not=f(g(x))\).

Example 2
Determining Whether Compositions of Functions is Commutative
Using the functions provided, find f(g(x)) and g(f(x)). Determine whether the
composition of the functions is commutative.
\(f(x)=2x+1\) and \(g(x)=3-x\)

Solution:
Let's begin by substituting g(x) into f(x).
\(f(g(x))=2(3-x)+1 = 6-2x+1 = 7-2x\)

Now, we can substitute f(x) into g(x).
\(g(f(x))=3-(2x+1) = 3-2x-1 = -2x+2\)

We find that \(g(f(x)) \not= f(g(x))\), so the operation of function composition
is not commutative.

Example 3
Interpreting Composite Functions
The function c(s) gives the number of calories burned completing s sit-ups, and
s(t) gives the number of sit-ups a person can complete in t minutes. Interpret
c(s(3)).

Solution:
The inside expression in the composition is s(3). Because the input to the
s-function is time, t=3 represents 3 minutes, and s(3) is the number of sit-ups
completed in 3 minutes.

Using s(3) as the input to the function c(s) gives us the number of calories
burned during the number of sit-ups that can be completed in 3 minutes, or
simply the number of calories burned in 3 minutes by doing sit-ups. 

Example 4
Investigating the Order of Function Composition
Suppose f(x) gives miles that can be driven in x hours and g(y) gives the
gallons of gas used in driving y miles. Which of the expressions is meaningful,
f(g(y)) or g(f(x))?

Solution:
The function \(y=f(x)\) is a function whose output is the number of miles driven
corresponding to the number of hours driven.
number of miles=f(number of hours)
The function g(y) is a function whose output is the number of gallons used
corresponding to the number of miles driven. This means:
number of gallons = g(number of miles)

The expression g(y) takes miles as the input and a number of gallons as the
output. The function f(x) requires a number of hours as the input. Trying to
input a number of gallons does not make sense. The expression f(g(y)) is
meaningless. 

The expression f(x) takes hours as the input and a number of miles driven as the
output. The function g(y) requires a number of miles as the input. Using
f(x)(miles driven) as an input value for g(y), where gallons of gas depends on
miles driven, does make sense. The expression g(f(x)) makes sense, and will
yield the number of gallons of gas used, g, driving a certain number of miles,
f(x), in x hours.

Are there any situations where f(g(y)) and g(f(x)) would both be meaningful or
useful expressions?
Yes. For many pure mathematical functions, both compositions make sense, even
though they usually produce different new functions. In real-world problems,
functions whose inputs and outputs have the same units also may give
compositions that are meaningful in either order.

Evaluating Composite Functions
Once we compose a new function from two existing functions, we need to be able
to evaluate it for any input in its domain. We will do this with specific
numerical inputs for functions expressed as tables, graphs, and formulas and
with variables as inputs to functions expressed as formulas. In each case, we
evaluate the inner function using the starting input and then use the inner
function's output as the input for the outer function.

When working with functions given as tables, we read input and output values
from the table entries and always work from the inside to the outside. We
evaluate the inside function first and then use the output as of the inside
function as the input to the outside function.

Example 5
Using a Table to Evaluate a Composite Function
Evaluate f(g(3)) and g(f(3)).

x     f(x)     g(x)
1     6        3
2     8        5
3     3        2
4     1        7

Solution:
To evaluate f(g(3)), we start from the inside with the input value 3. We then
evaluate the inside expression g(3) using the table that defines the function
g:g(3)=2. We can then use that result as the input to the function f, sp g(3) is
replaced by 2 and we get f(2). Then using the table that defines the function f,
we find that f(2)=8.
\(g(3)=2\)
\(f(g(3))=f(2)=8\)

To evaluate g(f(3)), we first evaluate the inside expression f(3) using the
first table: f(3)=3. Then, using the table for g, we can evaluate:
\(g(f(3))=g(3)-2\)

Evaluating Composite Functions Using Graphs
When we are given individual functions as graphs, the procedure for evaluating
composite functions is similar to the process we use for evaluating tables. We
read the input and output values, but this time, from the x and y axes of the
graphs.

Given a composite function and graphs of its individual functions, evaluate it
using the information provided by the graphs.
1. Locate the given input to the inner function on the x-axis of its graph
1. Read off the input of the inner function from the y-axis of its graph
3. Locate the inner function output on the x-axis of the graph of the outer
function.
4. Read the output of the outer function from the y-axis of its graph. This is
the output of the composite function.

Example 6
Using a Graph to Evaluate a Composite Function
Evaluate f(g(1))

Solution:
We evaluate g(1) using the graph of g(x), finding the input of 1 on the x-axis
and finding the output value of the graph at that input. Here, g(1)=3. We
use this value as the input to the function f.
\(f(g(1))=f(3)\)
We can then evaluate the composite function by looking to the graph of f(x),
finding the input of 3 on the x-axis and reading the output value of the graph
at this input. Here, \(f(3)=6, \text{ so } f(g(1))=6\).

Evaluating Composite Functions Using Formulas
When evaluating a composite function where we have either created or been
given formulas, the rule of working from the inside out remains the same. The
input value to the outer function will be the output of the inner function,
which may be a numerical value, a variable name, or a more complicated
expression. 

While we can compose the functions for each individual input value, it is
sometimes helpful to find a single formula that will calculate the result of a
composition \(f9g(x))\). To do this, we will extend our idea of the function
evaluation. Recall that, when we evaluate a function like \(f(t)=t^2-t\), we
substitute the value inside the parentheses into the formula wherever we see
the input variable. 

Given a formula for a composite function, evaluate the function.
1. Evaluate the inside out function using the input value or variable
provided.
2. use the resulting output as the input to the outside function.

Example 7
Evaluating a Composition of Functions Expressed as Formulas with a numerical
Input
Given \(f(t)=t^2-t\) and \(h(x)=3x+2\), evaluate f(h(1)).

Solution:
Because the inside expression is h(1), we start by evaluating h(x) at 1.
\(h(1)=3(1)+2 = 5\)
Then \(f(h(1))=f(5)\), so we evaluate f9t) at an input of 5.
\(f(h(1))= f(5) = 5^2-5 = 20\)

It makes no difference what the input variables t and x were called because we
evaluated for specific numerical values.

Finding the Domain of a Composite Function
As we discussed previously, the domain of a composite function such as \(f
\circ g\) is dependent on the domain of g and the domain of f. It is
important to know when we can apply a composite function and when we
cannot, that is, to know the domain of a function such as \(f \circ g\). Let us
assume we know the domains of the functions f and g separately. If we write
the composite function for an input x as f(g(x)), we can see right away that x
must be a member of the domain of g in order for the expression to be
meaningful, because otherwise we cannot complete the inner function
evaluation. However, we can see right away that x must be a member of the domain
of f, otherwise the second function evaluation in f(g(x)) cannot be
completed, and the expression is still undefined. Thus the domain of \(f \circ
g\) consists of only those inputs in the domain of g that produce outputs from
g belonging to the domain of f. Note that the domain of f composed with g is
the set of all x such that x is in the domain of f composed with g is the
set of all x such that x is in the domain of g and g9x) is in the domain of
f.

Domain of a Composite Function
The domain of a composite function f(g(x)) is the set of those inputs x in the
domain of g for which g(x) is in the domain of f.

Given a function composition f(g(x)), determine its domain
1. Find the domain of g
2. Find the domain of f
3. Find those inputs x in the domain of g for which g(x) is in the domain of
f. That is, exclude those inputs x from the domain of g for which g(x) is not
in the domain of f. The resulting set is the domain of \(f \circ g\).

Example 8
Finding the Domain of a Composite Function
Find the domain of \((f \circ g)(x)\) where \(f(x)=5/(x-1) \text{ and } g(x) =
4/(3x-2)\).

Solution:
The domain of g(x) consists of all real numbers except \(x=2/3\), since that
input value would cause us to divide by 0. Likewise, the domain of f consists
of all real numbers except 1. So we need to exclude from the domain of g(x)
that value of x for which g9x)=1.

\(\frac{4}{3x-2} = 1\)
\(4=3x-2\)
\(6=3x = 2\)

So the domain of \(f \circ g\) is the set of all real numbers except 2/3 and 2.
This means that :
\(x \not= \frac{2}{3} \text{ or } x\not=2\)
We can write this in interval notation as:
\((-\infty,\frac{2}{3}) \cup (\frac{2}{3},2) \cup (2,\infty)\)

Example 9
Finding the Domain of a Composite Function Involving Radicals
Find the domain of :
\((f \circ g)(x) \text{ where } f(x) = \sqrt{x+2} \text{ and } g(x) =
\sqrt{3-x}\)

Solution:
Because we cannot take the square root of a negative number, the domain of g
is \((-\infty,3]\). Now we can check the domain of the composite function:
\((f \circ g)(x) = \sqrt{\sqrt{3-x}+2}\)
This expression must \(\geq 0\), since the radicand of a square root must be
positive. Since square roots are positive, \(\sqrt{3-x} \geq 0\) which gives
the domain of \((-\infty,3]\).

This example shows that knowledge of the range of functions (specifically
the inner function) can also be helpful in finding the domain of a composite
function. It also shows that the domain of \(f \circ g\) can contain values
that are not in the domain of f, though they must be in the domain of g.

Decomposing a Composite Function into its Component Functions
In some cases, it is necessary to decompose a complicated function. In
other words, we can write it as a composition of two simpler functions. There
may be more than one way to decompose a composite function, so we may
choose the decomposition that appears to be most expedient.

Example 10
Decomposing a Function
Write \(f(x)=\sqrt{5-x^2}\) as the composition of two functions.

We are looking for two functions, g and h, so f(x)=g(h(x)). To do this,
we look for a function inside a function in the formula for f(x). As one
possibility, we might notice that the expression \(5-x^2\) is the inside of
the square root. We could then decompose the function as \(h(x)=5-x^2 \text{
and } g(x) = \sqrt{x}\)
We can check out answer by recomposing the functions.
\(g(h(x))=g(5-x^2)=\sqrt{5-x^2}\)

Transformations of Functions

Graphing Functions Using Vertical and Horizontal Shifts 
Often, when given a problem, we try to model the scenario using
mathematics in the form of words, tables, graphs, and equations. One method we
can employ is to adapt the basic graphs of the toolkit functions to build new
models for a given scenario. There are systematic ways to alter functions to
construct appropriate models for the problems we are trying to solve.

Identifying Vertical Shifts
One simple kind of transformation involves shifting the entire graph of a
function up, down, right, or left. The simplest shift is a vertical shift,
moving the graph up or down, because this transformation involves
adding a positive or negative constant to the function. In other words,
we add the same constant to the output value of the function regardless of
the input. For a function \(g(x)=f(x)+k\), the function f(x) is shifted
vertically k units. To help visualize the concept of a vertical shift, consider
that \(y=f(x)\). Therefore, \(f(x)+k\) is equivalent to \(y+k\). Every unit of
y is replaced by \(y+k\), so the y-value increases or decreases depending on
the value of k. The result is a shift upward or downward.

Vertical Shift
Given a function, f(x), a new function g(x)=f(x)+k where k is a constant, is
a vertical shift of the function f(x). All the output values change by k units.
If k is positive, the graph will shift up. If k is negative, the graph will
shift down. 

Example 1
Adding a Constant to a Function
To regulate temperature in a green building, airflow vents near the roof open
and close throughout the day. During the summer, the facilities manager
decides to try to better regulate temperature by increasing the amount of
open vents by 20 square feet throughout the day and night. Sketch a graph of
this new function. 

Solution:
We can sketch a graph of this new function by adding 20 to each of the output
values of the original function. This will have the effect of shifting the
graph vertically up. Notice that for each input value, the output value has
increased by 20, so if we call the new function S(t), we could write
\(S(t)=V(t)+20\).

This notation tells us that, for any value of t, S(t) can be found by
evaluating the function V at the same input and then adding 20 to the result.
This defines S as a transformation of the function V, in this case a vertical
shift up 20 units. Notice that, with a vertical shift, the input values stay the
same and only the output values change.

Given a tabular function, create a new row to represent a vertical
shift.
1. Identify the output row or column
2. Determine the magnitude of the shift
3. Add the shift to the value in each output cell. Add a positive value for up
or a negative value for down.

Example 2
Shifting a Tabular Function Vertically

Solution:
The formula g(x)=f(x)-3 tells us that we can find the output values of g by
subtracting 3 from the output values of f.
\(f(2)=1\)
\(g(x)=f(x)-3\)
\(g(2)=f(2)-3\)
\(1-3=-2\)

As with the earlier vertical shift, notice the input values stay the same and
only the output values change.

Identifying Horizontal Shifts
We just saw that the vertical shift is a change to the output, or outside of the
function. We will now look at how changes to input, on the inside of the
function, change its graph and meaning. A shift to the input results in a
movement of the graph of the function left or right in what is known as a
horizontal shift.

For example, if \(f(x)=x^2\). then \(g(x)=(x-2)^2\) is a new function. Each
input is reduced by 2 prior to squaring the function. The result is that the
graph is shifted 2 units to the right, because we would need to increase
the prior input by 2 units to yield the same output value as given in f.

Horizontal Shift
Given a function, f, a new function g(x)=f(x-h), where h is a constant, is a
horizontal shift of the function f. If h is positive the graph will shift to
the right. If h is negative then the graph will shift to the left. 

Example 3
Adding a Constant to an Input
Returning to our building airflow example, suppose that in autumn the
facilities manager decides that the original venting plan starts too late, and
wants to begin the entire venting program 2 hours earlier. Sketch a graph
of the new function.

Solution:
We can set V(t) to be the original program and F(t) to be the revised program.
V(t)= the original venting plan
F(t)= starting 2 hours sooner

in the new graph, at each time, the airflow is the same as the original
function V, was 2 hours later. For example, in the original function V, the
airflow starts to change at 8 am, whereas for the function F, the airflow
starts to change at 6 am. The comparable function values are V(8)=F=(6).
Notice also that the vents first opened top \(220 ft^2\) at 10 am under the
original plan, while under the new plan the vents reach \(220 ft^2\) at
8am, so V(10)=F(8). In both cases, we see that,. because F(t) starts 2
hours sooner, h=-2. That means that the same output values are reached when:
\(F(t)=V(t)-(-2)) = V(t+2)\)

Note that V(t+2) has the effect of shifting the graph to the left. Horizontal
changes affect the domain of a function instead of the range and often seem
counterintuitive. The new function F(t) uses the same outputs as V(t), but
matches those outputs to inputs 2 hours earlier than those of V(t). Said
another way, we must add 2 hours to the input of V to find the corresponding
output of F: F(t)=V(t+2).

Given a tabular function, create a new row to represent a horizontal
shift.
1. Identify the input row or column
2. Determine the magnitude of the shift
3. Add the shift to the value in each input cell.

Example 4
Shifting a Tabular Function Horizontally
A function f(x) is given, create a table for the function g(x)=f(x-3).

Solution:
The formula \(g(x)=f(x-3)\) tells us that the output values of g are the same
as the output value of f when the input value is 3 less than the original
value. for example, we know that f(2)=1. To get the same output from the
function g, we will need an input value that is 3 larger. We input a value
that is 3 larger for g(x) because the function takes 3 away before evaluating
the function f.
\(g(5)=f(5-3)=f(2)=1\)

The result is that the function g(x) has been shifted to the right by 3. Notice
the output values for g(x) remain the same as the output values for f(x),
but the corresponding values, x, have shifted to the right by 3. Specifically,
2 shifted to 5,4 shifted to 7, 6 shifted to 9, and 8 shifted to 11.

Example 5
Identifying a Horizontal Shift of a Toolkit function
The graph represents a transformation of the toolkit function \(f(x)=x^2\).
Relate this new function g(x) to f(x), and then find a formula for g(x).

Solution:
Notice that the graph is identical in shape to the \(f(x)=x^2\) function,
but the x-values are shifted to the right 2 units. The vertex used to be at
(0,0), but now the vertex is at (2,0). The graph is the basic quadratic
function shifted 2 units to the right.
\(g(x)=f(x-2)\)

Notice how we must input the value x=2 to get the output value y=0. The
x-values must be 2 units larger because of the shift to the right by 2 units.
We can then use the definition of the f(x) function to write a formula for g(x)
by evaluating f(x-2).
\(F(x)=x^2\)
\(g(x)=f(x-2)\)
\(g(x)=f(x)-20=(x-2)^2\)

To determine whether the shift is +2 or -2, consider a single reference point on
the graph. For a quadratic, looking at the vertex point is convenient. In
the original function, f(0)=0. In out shifted function, g(2)=0. To
obtain the output value 0 from the function f, we need to decide whether a plus
or minus sign will work to satisfy g(2)=f(x-2)=f(0)=0. For this to work, we will
need to subtract 2 units form out input values.

Example 6
Interpreting Horizontal versus Vertical Shifts
The function G(m) gives the number of gallons of gas required to drive m
miles. Interpret G(m)+10 and G(m+10).

Solution:
G(m)+10 can be interpreted as adding 10 to the output, gallons. This is the gas
required to drive m miles, plus another 10 gallons of gas. The graph would
indicate a vertical shift.

G(m+10) can be interpreted as adding 10 to the input, miles. So this is the
number of gallons of gas required to drive 10 miles more than m miles. The
graph would indicate a horizontal shift.

Combining Vertical and Horizontal Shifts
Now that we have two transformations, we can combine them together. Vertical
shifts are outside changes that affect the input x axis values and shift
the function left or right. Combining the two types of shifts will cause the
graph of a function to shift up or down and right or left.

Given a function and both a vertical and a horizontal shift, sketch the graph.
1. Identify the vertical and horizontal shifts from the formula.
2. The vertical shift results from a constant added to the output. Move the
graph up for a positive constant and down for a negative constant.
3. The horizontal shift results from a constant added to the input. Move the
graph left for a positive constant and right for a negative constant. 
4. Apply the shifts to the graph in either order.

Example 7
Graphing Combined Vertical and Horizontal Shifts
Given f(x)=|x|, sketch a graph of h(x)=f(x+1)-3

Solution:
The function f is our toolkit absolute value function. We know that this graph
has a v shape, with the point at the origin. The graph of h has transformed f in
two ways: f(x+1) is a change on the inside of the function, giving a
horizontal shift left by 1, and the subtraction by 3 in f(x+1)-3 is a change
to the outside of the function, giving a vertical shift down by 3.

Example 8
Identifying Combined Vertical and Horizontal Shifts
Write a formula for the graph shown, which is a transformation of the
toolkit square root function.

Solution:
The graph of the toolkit function starts at the origin, so this graph has
been shifted 1 to the right and up 2. In function notation, we could write
that as \(h(x)=f(x-2)+2\). Using the formula for the square root function, we
can write \(h(x)=\sqrt{x-1}+2\).

Note that this transformation has changed the domain and range of the
function. This new graph has domain \([1,\infty)\) and range \([2,\infty)\).

Graphing Functions Using Reflections about the Axes
Notice that the vertical reflection produces a new graph that is a mirror
image of the base or original graph about the y-axis. The horizontal reflection
produces a new graph that is a mirror image of the base or original graph
about the y-axis.

Reflections
Given a function f(x), a new function g(x)=-f(x) is a vertical reflection of
the function f(x), sometimes called a reflection about the x-axis.
Given a function f(x), a new function g(x)=f(-x) is a horizontal reflection of
the function f(x), sometimes called a reflection about the y-axis.

Given a function, reflect the graph both vertically and horizontally
1. Multiply all outputs by -1 for a vertical reflection. The new graph
is a reflection of the original graph about the x-axis.
2. Multiply all outputs by -1 for a horizontal reflection. The new graph
is a reflection of the original graph about the y-axis.

Example 9
Reflecting a Graph Horizontally and Vertically
Reflect the graph of \(s(t)=\sqrt{t}\) vertically and horizontally.

Solution:
Reflecting the graph vertically means that each output value will be reflected
over the horizontal t-axis.
Because each output value is the opposite of the original output value, we
can write: \(V(t)=-s(t)\).
Notice that his is an outside change, or vertical shift, that affects the
output s(t) values, so the negative sign belongs outside of the function.
Reflecting horizontally means that each input value will be reflected over
the vertical axis.
Because each input value is the opposite of the original input value, we
can write: \(H(t)=s(-t)\).
Notice that this is an inside change or horizontal change that affects the
input values, so the negative sign is on the inside of the function.

Note that these transformation can affect the domain and range of the
functions. While the original square root function has domain \([0,\infty)\) and
range \([0,\infty)\), the vertical reflection gives the V(t) function the range
\((-\infty.0])\) and the horizontal reflection gives the H(t) function the
domain \((-\infty,0]\).

Example 10
Reflecting a Tabular Function Horizontally and Vertically
A function f(x) is given, create a table for the functions:
g(x)=-f(x)
h(x)=f(-x)
x   2   4   6   8
f(x)1   3   7   11

Solution:
For g(x), the negative sign outside the function indicates a vertical
reflection, so the x-values stay the same and each output value will be the
opposite of the original output value.

For h(x), the negative sign inside the function indicates a horizontal
reflection, so each input value will be the original input value and the h(x)
values stay the same as the f(x) values.

Example 11
Applying a Learning Model Equation
A common model for learning has an equation similar to \(k(t)=2^{-2} + 1\),
where k is the percentage of mastery that can be achieved after t
practice sessions. This is a transformation of the function \(f(t)=2^t\).

Solution:
This equation combines three transformations into one equation.
1. A horizontal reflection:\(f(-t)=2^{-t}\)
2. A vertical reflection:\(-f(-t)=-2^{-t}\)
3. A vertical shift:\(-f(-t)=-2^{-t} + 1\)

We can sketch a graph by applying these transformations one at a
time to the original function. Let us follow two points through each of
the three transformations. We will choose the points (0,1) and (1,2).
1. First, we apply a horizontal reflection: (0,1)(-1,2)
2. Then, we apply a vertical reflection:(0,-1)(-1,-2)
3. Finally, we apply a vertical shift:(0,0)(-1,-1)

This means that the original points, (0,1) and (1,2) become (0,0) and (-1,-1)
after we apply the transformations.

As a model for learning, this function would be limited to a domain of
\(t \geq 0\), with a corresponding range of [0,1).

Determining Even and Odd Functions
Some functions exhibit symmetry so that reflections result in the original
graph. For example, horizontally reflecting the toolkit functions
\(f(x)=x^2\) or \(f(x)=|x|\) will result in the original graph. We say
these types of graphs are symmetric about the y-axis. Functions whose graphs
are symmetric about the y-axis are called even functions. In the graphs
of \(f(x)=x^3\) or \(f(x)=1/x\) were reflected over both axes, the result
would be the original graph.

We say that these graphs are symmetric about the origin. A function with a
graph that is symmetric about the origin is called an odd function. A function
can be neither even nor odd if it does not exhibit symmetry. For example,
\(f(x)=2^x\) is neither even nor odd. Also, the only function that is both
even and odd is the constant function \(f(x)=0\).

Even and Odd Functions
A function is called an even function if for every input of x:
\(f(x)=f(-x)\).
The graph of an even function is symmetric about the y-axis.
A function is called an odd function if for every input of x:
\(f(x)=-f(-x)\).
The graph of an odd function is symmetric about the origin.

Given the formula for a function, determine if the function is even, odd, or
neither.
1. Determine whether the function satisfies \(f(x)=f(-x)\). If it does, it is
even.
2. Determine whether the function satisfies \(f(x)=-f(-x)\). If it does, it
is odd. 
3. If the function does not satisfy either rule, it is neither even nor odd.

Example 12
Determining whether a Function is Even, odd, or neither
Is the function \(f(x)=x^3+2x\) even, odd, or neither?

Solution:
Without looking at a graph, we can determine whether the function is even or
odd by finding formulas for the reflections and determining if they
return us to the original function. Let us begin with the rule for even
functions.
\(f(-x)=(-x)^3 + 2(-x) = -x^3 - 2x\)
This does not return us to the original function, so this function is not
even. We can now test the rule for odd functions.
\(-f(-x)=-(-x^3-2x)=x^3+2x\)
Because \(-f(-x)=f(x)\), this is an odd function.

Consider the graph of f. Notice that the graph is symmetric about the
origin. For every point (x,y) on the graph, the corresponding
point(-x,-y) is also on the graph. For example, (1,3) is on the graph of f, and
the corresponding point (-1,-3) is also on the graph.

Graphing Functions Using Stretches and Compressions
Adding a constant to the inputs of a function changed the position of a graph
with respect to the axes, but it did not affect the shape of a graph. We now
explore the effects of multiplying the inputs or outputs by some quantity.
We can transform the inside (input values) of a function or we can
transform the outside (output values) of a function. Each change has a
specific effect that can be seen graphically.

Vertical Stretches and Compressions
When we multiply a function by a positive constant, we get a function whose
graph is stretched or compressed vertically in relation to the graph of
the original function. If the constant is greater than 1, we get a vertical
stretch. If the constant is between 0 and 1, we get a vertical compression. 

Vertical Stretches and Compressions
Given a function f(x), a new function g(x)=af(x), where a is a constant, is
a vertical stretch or vertical compression of the function f(x).
1. If a>1 then the graph will be stretched.
2. If 0<a<1, then the graph will be compressed.
3. If a<0, then there will be a combination of a vertical stretch with a
vertical reflection.

Given a function, graph its vertical stretch
1. Identify the value of a
2. Multiply all range values by a
3. If a>1, the graph is stretched by a factor of a
   If 0<a(1, the graph is compressed by a  factor of a
   If a<0, the graph is either stretched or compressed and also reflected about
the x-axis.

Example 13
Graphing a Vertical Stretch
A function P(t) models the population of fruit flies.
A scientist is comparing this population to another population, Q, whose
growth follows the same pattern, but is twice as large. Sketch a graph of
this population.

Solution:
Because the population is always twice as large, the new population's output
values are always twice the original function's output values. 
If we choose four reference points, (0,1)(3,3)(6,2)(7,0), we will multiply all
of the outputs by 2.
The following shows where the new points for the new graph will be located.

This means that for any input t, the value of the function Q is twice the
value of the function P. Notice that the effect on the graph is a vertical
stretching of the graph, where every point doubles its distance from the
horizontal axis. The input values, t, stay the same while the output values
are twice as large as before. 

Given a tabular function and assuming that the transformation is a
vertical stretch or compression, create a table for a vertical compression.
1. Determine the value of a
2. Multiply all of the output values by a

Example 14
Finding a Vertical Compression of a Tabular Function
A function f is given, create a table for the function \(g(x)=\frac{1}{2}f(x)\).
  x     2     4     6     8
f(x)    1     3     7     11

Solution:
The formula \(g(x)=\frac{1}{2}f(x)\) tells us that the output values of g are
half of the output values of f with the same inputs. For example, we know that
f(4)=3. Then
\(g(4)=\frac{1}{2}f(4)=\frac{1}{2}(3)=\frac{3}{2}\)

The result is that the function g(x) has been compressed vertically by
1/2. Each output value is divided in half, so the graph is half the
original height.

Example 15
Recognizing a Vertical Stretch
The graph shown is a transformation of the toolkit function \(f(x)=x^3\).
Relate this new function g(x) to f(x), and then find a formula for g(x).

When trying to determine a vertical stretch or shift, it is helpful to look for
a point on the graph that is relatively clear. in this graph, it appears that
g(2)=2. With the basic cubic function at the same input, \(f(2)=2^3=8\).
based on that, it appears that the outputs of g are 1/4 the outputs of the
function f because g(2)=1/4 f92). From this we can safely conclude that
g9x)=1/4f(x). We can write a formula for g by using the definition of f.
\(g(x)=\frac{1}{4}f(x)=\frac{1}{4}x^3\)

Horizontal Stretches and Compressions
Now we consider changes to the inside of the function. When we multiply a
function's input by a positive constant, we get a function whose graph is
stretched or compressed horizontally in relation to the graph of the original
function. If the constant is between 0 and 1, we get a horizontal stretch; if
the constant is greater than 1, we get a horizontal compression of the function.

Given a function y=f(x), the form y=f(bx) results in a horizontal stretch or
compression. Consider the function \(y=x^2\). The graph of \(y=(0.5x)^2\) is a
horizontal stretch of the graph of the function \(y=x^2\) by a factor of 1/2.

Horizontal Stretches and Compressions
Given a function f(x), a new function g(x)=f(bx), where b is a constant, and is
a horizontal stretch or horizontal compression of the function f(x).
1. If b>1, then the graph will be compressed 1/b.
2. If 0<b<1, then the graph will be stretched by 1/b.
3. If b<0<1, then there will be combination of a horizontal stretch or
horizontal compression with a horizontal reflection.

Given a description of a function, sketch a horizontal compression or stretch.
1. Write a formula to represent the function
2. Set g(x)=f(bx) where b>1 for a compression or 0<b<1 for a stretch.

Example 16
Graphing a Horizontal Compression
Suppose a scientist is comparing a population of fruit flies to a population
that progresses through its lifespan twice as fast as the original population.
In other words, this new population will progress in 1 hour the same amount as
the original population does in 2 hours, and in 2 hours, it will progress as
much as the original population does in 4 hours. Sketch a graph of this
population.

Symbolically, we could write:
\(R(1)=P(2)\)
\(R(2)=P(4)\)
\(R(t)=P(2t)\)

Note that the effect on the graph is a horizontal compression where all input
values are half of their original distance from the vertical axis.

Example 17
Finding a Horizontal Stretch for a Tabular Function
A function f(x) is given, create a table for the function g(x)=f(1/2x).
 x     2     4     6     8
f(x)   1     3     7     11

Solution:
The formula g(x)=f(1/2x) tells us that the output values for g are the same as
the output values for the function f at an input half the size. Notice that we
do not have enough information to determine g(2) because g(2)-f(1/2(2))=f(1),
and we do not have a value for f(1) in our table. Our input values to g will
need to be twice as large to get inputs for f that we can evaluate. For example,
we can determine g(4).
\(g(4)=f(1/2(4))=f(2)=1\)

Because each input value has been doubled, the result is that the function g(x)
has been stretched horizontally by a factor of 2.

Example 18
Recognizing a Horizontal Compression on a Graph
Relate the function g(x) to f(x).

Solution:
The graph of g(x) looks like the graph of f(x) horizontally compressed. Because
f(x) ends at (6,4) and g(x) ends at (2,4), we can see that the x-values have
been compressed to 1/3, because 6(1/3)=2. We might also notice that g(2)=f(6)
and g(1)=f(3). Either way, we can describe this relationship as g(x)=f(3x). This
is a horizontal compression by 1/3.

Notice that the coefficient needed for a horizontal stretch or compression is
the reciprocal of the stretch or compression. So, to stretch the graph
horizontally by a scale factor of 4, we need a coefficient of 1/4 in our
function: f(1/4(x)). This means that the input values must be 4 times larger to
produce the same result, requiring the input to be larger, causing the
horizontal stretching.

Performing a Sequence of Transformations
When combining transformations, it is very important to consider the order of
the transformations. For example, vertically shifting by 3 and then vertically
stretching by 2 does not create the same graph as vertically stretching by 2 and
then vertically shifting by 3, because when we shift first, both the original
function and the shift get stretched, while only the original function gets
stretched when we stretch first.

When we see an expression such as 2f(x)+3, which transformation should we start
with? The answer here follows nicely from the order of operations. Given the
output value of f(x), we first multiply by 2, causing the vertical stretch, and
then add 3, causing the vertical shift. In other words, multiplication before
addition.

Horizontal transformations are a little trickier to think about. When we write
g(x)=f(2x+3), for example, we have to think about how the inputs to the function
g relate to the inputs to the function f. Suppose we know f(7)=12. What input to
g would produce that output? In other words, what value of x will allow
g(x)=f(2x+3)=12? We would need 2x+3=7. To solve for x, we would first subtract
3, resulting in a horizontal shift, and then divide by 2, causing a horizontal
compression. 

This format ends up being very difficult to work with, because it is usually
much easier to horizontally stretch a graph before shifting. We can work around
this by factoring inside the function.
\(f(bx+p)=f(b(x+\frac{p}{b}))\)
So:
\(f(x)=(2x+4)^2\)
We can factor out a 2
\(f(x)=(2(x+2))^2\)
Now we can more clearly observe a horizontal shift to the left 2 units and a
horizontal compression. Factoring in this way allows us to horizontally stretch
first and then shift horizontally.

Combining Transformations
When combining vertical transformations written in the form af(x)+k, first
vertically stretch by a and then vertically shift by k.

When combining horizontal transformations written in the form f(bx-h), first
horizontally shift by h/b and then horizontally stretch by 1/b.

When combining horizontal transformations written in the form f(b(x-h)), first
horizontally stretch by 1/b and then horizontally shift by h.

Horizontal and vertical transformation are independent. It does not matter
whether horizontal or vertical transformations are performed first.

Example 19
Finding a Triple Transformation of a Tabular Function
For the function f(x), create a table of values for the function g(x)=2f(3x)+1.
  x     6     12     18     24
f(x)    10    14     15     17

Solution:
There are three steps to this transformation, and we will work from the inside
out. Starting with the horizontal transformations, f(3x) is a horizontal
compression by 1/3, which means we multiply each x-value by 1/3.

   x     2     4     6     8
f(x)     10   14    15    17

Looking now to the vertical transformations, we start with the vertical
stretch, which will multiply the output values by 2. We apply this to the
previous transformation.

   x     2     4     6     8
2f(3x)  20    28    30    34

Finally, we can apply the vertical shift, which will add 1 to all the output
values.

x                 2     4     6     8
g(x)=2f(3x)+1     21    29    31    35

Example 20
Finding a Triple Transformation of a Graph
Use the graph of f(x) to sketch a graph of k(x)=f(1/2x+1)-3

Solution:
To simplify, let us start by factoring out the inside of the function.
\(f(\frac({1}{2}x+1)-3=f(\frac{1}{2}(x+2)-3\)

By factoring the inside, we can first horizontally stretch by 2, as indicated by
the 1/2 on the inside of the function. Remember that twice the size of 0 is
still 0, so the point (0,2) remains at (0,2) while the point (2,0) will stretch
to (4,0).

Next, we horizontally shift left by 2 units, as indicated by x+2.

Last, we vertically shift down by 3 to complete our sketch, as indicated by the
-3 on the outside of the function.

Quadratic Models

When a mathematical model leads to a quadratic function, the properties of the
graph of the quadratic function can provide important information about the
model. We can use the quadratic function to determine the maximum or minimum
value of the function. The fact that the graph of a quadratic function has a
maximum or minimum value enables us to answer questions involving optimization
or finding the maximum or minimum values involving quadratic functions.

Solving Applied Problems
In economics, revenue is defined as the amount of money received from the sale
of an item and is equal to the unit selling price of the item times the number
of units that were sold.

\(R=xp\)

In economics, the Law of Demand states that p and x are related. As one
increases, the other decreases. The equation that relates p and x is called the
demand equation.

Example 1
The marketing department at Texas Instruments has found that, when certain
calculators are sold at a certain price, the number of calculators sold is given
by the demand equation:

\(x=21000-150p\)

1. Express the revenue as a function of price 
2. What unit price should be used to maximize revenue
3. If this price is charged, what is the maximum revenue
4. How many units are sold at this price
5. Graph

Solution:
1. The revenue is \(R=xp\) where \(x=21000-150p\).
   \((21000-150p)p= -150p^2+21000p\)

2. The function is a quadratic function with a=-150, b=21000, and c=0.
   Because a<0, the vertex is the highest point on the parabola.
   The revenue is therefore a maximum when the price is:
   \(p=\frac{-b}{2a} = -\frac{21000}{2(-159)} = \frac{21000}{-300} = $70.00\)

3. The maximum revenue is: 
   \(R(70) = -150(70)^2 + 21000(70) = $735000\)

4. The number of calculators sold is given by the demand equation:
   \(x=21000-150p\)
   At a price of \(p=$70\),
   \(x=21000-150(70) = 10500\)

Example 2
Maximizing the Area Enclosed by a Fence
A farmer has 2000 yards of fence to enclose a rectangular field. What are the
dimensions of the rectangle that encloses the most area?

Solution:
The available fence represents the perimeter of the rectangle. If x is the
length and w is the width, then:
\(2x+2w=2000\)
The area of the rectangle is:
\(a=xw\)
To express area in terms of a single variable, we solve the equation for w and
substitute the result in \(a=xw\). Then area involves only the variable x. You
could also solve the equation for x and express area in terms of w alone.
\(2x + 2w = 2000\)
\(2w = 2000-2x\)
\(w = \frac{2000-2x}{2} = (1000-x)\)
So, the area is:
\(a = xw = x(1000-x) = -x^2 + 1000x\)
Now, area is a quadratic function of x.
\(a(x)=-x^2+1000x\)
When you graph this, you see a<0, so the vertex is a maximum point on the graph.
The maximum value occurs at:
\(x=-\frac{b}{2a} = -\frac{1000}{2(-1)} = 500\)
The maximum value of a is:
\(a(-\frac{b}{2a}) = a(500) = -500^2 + 1000(500) = -250000 + 500000 = 250000\)
The largest rectangle that can be enclosed by 2000 yards of fence has an area of
250000 square yards. Its dimensions are 500 by 500 yards.

Example 3
Analyzing the motion of a projectile
A projectile is fired from a cliff 500 feet above the water at an inclination of
45 degrees to the horizontal, with a muzzle velocity of 400 feet per second. In
physics, it is established that the height of the projectile above the water is
given by:

\(h(x)=\frac{-32x^2}{(400)^2} + x + 500\)

X is the horizontal distance  of the projectile from the base of the cliff.
1. Find the maximum height of the projectile
2. How far from the base of the cliff will the projectile strike the water?

Solution:
1. The height of the projectile is given by a quadratic function.
   \(h(x)=\frac{-32x^2}{(400)^2}+x+500=\frac{-1}{5000}x^2+x+500\)
   We are looking for the maximum value of h. Since a<0, the maximum value is
   obtained at the vertex.
   \(x=-\frac{b}{2a}=-\frac{1}{2(-1/5000)}=\frac{5000}{2}=2500\)
   The maximum height of the projectile is:
   \(h(2500)=\frac{-1}{5000}(2500)^2+2500+500=-1250+2500+500=1750ft\)

2. The projectile will strike the water when the height is zero. To find the
   distance x traveled, we need to solve the equation:
   \(h(x)=\frac{-1}{5000}x^2+x+500=0\)
   We find the descriminant first.
   \(b^2-4ac=1^2-4(\frac{-1}{5000})(500)=1.4\)
   Then: \(x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}=\frac{-1 \pm
   \sqrt{1.4}}{2(-1/5000)}=5458ft\)

Inequalities with Quadratic Functions

In this section, we solve inequalities that involve quadratic functions. We will
accomplish this by using their graphs. For example, to solve the inequality
\(ax^2+bx+c>0\)
we graph the function \(f(x)=ax^2+bx+c\) and from the graph, determine where it
is above the x-axis and where f(x)>0. To solve the inequality, we graph the
function and determine where the graph is below the x-axis. If the inequality is
not strict, we include the x-intercepts in the solution.

Example 1
Solving an Inequality
Solve the inequality \(x^2-4x-12\leq0\) and graph the solution set.

Solution:
We graph the function \(f(x)=x^2-4x-12\)
The intercepts are: x=6 and x=-2
the y-intercept is -12 with the x-intercepts at 6 and -2.
The vertex is at: 
\(x=-\frac{b}{2a}=-\frac{-4}{2}=2\)
Since f(2)=-16, the vertex is (2,-16).
The graph is below the x-axis for -2<x<6. Since the original inequality is not
strict, we include the x-intercepts. The solution set is \({x|-2\leqx\leq6} or
[-2,6].

Example 2
Solving an Inequality
Solve the inequality \(2x^2<x+10\) and graph the solution set.

Solution:
We arrange the inequality so that 0 is on the right side.
\(2x^2<x+10\)
\(2x^2-x-10<0\)
This inequality is equivalent to the one that we wish to solve.
Next, we graph the function \(f(x)=2x^2-x-10\). The intercepts are:
y=-10, x=-2, x=5/2
The vertex is at \(x=-\frac{b}{2a}=-\frac{-1}{4}=\frac{1}{4}\)
Since f(1/4)=-10.125, the vertex is at \((\frac{1}{4},-10.125)\)
The graph is below the x-axis between x=-2 and x=5/2. Since the inequality is
strict, the solution set is \({x|-2<x<\frac{5}{2}}\) or \((-2,\frac{5}{2})\).

Polynomial Functions and Models

Polynomial Functions
We can write linear functions as \(f(x)=mx+b\) and quadratic functions as
\(f(x)=ax^2+bx+c\). Each of these functions are polynomial functions. The
domain of a polynomial function is the set of all real numbers. 

A polynomial function is a function whose rule is given by a polynomial in one
variable. The degree of a polynomial function is the largest power of x that
appears. The zero polynomial function \(f(x)=0\) is not assigned a degree.
Polynomial functions are among the simplest expressions in algebra. They are
easy to evaluate: only addition and multiplication are required. Because of
this, they are often used to approximate other, more complicated functions. 

Example 1
Identifying polynomial functions
Determine which of the following are polynomial functions. For those that are,
state the degree.
1. \(f(x)=2-3x^4\)
2. \(g(x)=\sqrt{x}\)
3. \(h(x)=\frac{x^2-2}{x^3-1}\)
4. \(f(x)=0\)
5. \(g(x)=8\)
6. \(h(x)=-2x^3(x-1)^2\)

Solution:
1. polynomial function of degree 4
2. not a polynomial function because the power is 1/2 which is not an integer
3. not a polynomial function. it is the ratio of two polynomials and the
   polynomial in the denominator is of positive degree.
4. polynomial function of degree zero, it is not assigned a degree
5. nonzero constant function, which is a polynomial function of degree 0
6. polynomial function of degree 5

You will learn that the graph of every polynomial function is both smooth and
continuous. By smooth, we mean that the graph contains no sharp corners and by
continuous, we mean that the graph has no gaps or holes and can be drawn without
lifting pencil from paper.

Power Functions
We begin the analysis of the graph of a polynomial function by discussing power
functions, a special kind of polynomial function. A power function of degree n
is a monomial of the form \(f(x)=ax^n\) where a is a real number, not zero, and
n>0.

Examples of power functions include:
\(f(x)=3x\) is degree 1
\(f(x)=-5x^2\) is degree 2
\(f(x)=8x^3\) is degree 3
\(f(x)=-5x^4\) is degree 4

The graph of a power function of degree 1, \(f(x)=ax\), is a straight line, with
slope a, that passes through the origin. The graph of a power function of degree
2, \(f(x)=ax^2\), is a parabola, with vertex at the origin, that opens up if a>0
and down if a<0.

If we know how to graph a power function of the form \(f(x)=x^n\), a compression
or stretch, and a reflection about the x-axis will enable us to obtain the graph
of \(g(x)=ax^n\). Consequently, we shall concentrate on graphing power functions
of the form \(f(x)=x^n\).

We begin with power functions of even degree of the form \(f(x)=x^n,n\geq2\) and
n even. The domain of f is the set of all real numbers, and the range is the set
of nonnegative real numbers. Such a power function is an even function, so its
graph is symmetric with respect to the y-axis. Its graph always contains the
origin and the points (-1,1) and (1,1).

If n=2, the graph is the familiar parabola \(y=x^2\) that opens up, with vertex
at the origin. If \(n\geq4\), the graph of \(f(x)=x^n\), n even, will be closer
to the x-axis than the parabola \(y=x^2\) if \(-1<x<1\) and farther from the
x-axis than the parabola \(y=x^2\) if \(x<-1\) or if \(x>1\). 

Properties of Power Functions
1. f is an even function, so its graph is symmetric with respect with respect to
   the y-axis.
2. The domain is the set of all real numbers. The range is the set of
   nonnegative real numbers.
3. The graph always contains the points (-1,1) and (1,1).
4. As the exponent n increases in magnitude, the graph becomes more vertical
   when x<-1 or x>1. But for x near the origin, the graph tends to flatten out   and lie closer to the x-axis.

Now we consider power functions of odd degree of the form \(f(x)=x^n,n\geq3\)
and n odd. The domain and range of f are the set of real numbers. Such a power
function is an odd function, so its graph is symmetric with respect to the
origin. Its graph always contains the points (-1,1) and (1,1).

The graph of \(f(x)=x^n\) when n=3 has been shown several times. If \(n\geq5\),
the graph of \(f(x)=x^n\), n odd, will be closer to the x-axis than that of
\(y=x^3\) if -1<x<1 and farther from the x-axis than that of \(y=x^3\) if x<-1
or if x>1.

Properties of Power Functions when x is Odd
1. f is an odd function, so its graph is symmetric with respect to the origin
2. The domain and the range are the set of all real numbers.
3. The graph always contains the points (-1,-1) and (1,1).
4. As the exponent n increases in magnitude, the graph becomes more vertical
   when x<-1 or x>1 but for x near the origin, the graph tends to flatten out
   and lie closer to the x-axis.

Graph Polynomial Functions using Transformation
The methods of shifting, compression, stretching, and reflection will enable us
to graph polynomial functions that are transformations of power functions.

Example 2
Graph \(f(x)=1-x^5\)

Solution:
First, start with the usual function of \(y=x^5\)
Multiply by -1 to reflect across the x-axis.
This gives us \(y=-x^5\).
Now, add 1 to equation which will shift the graph up 1 unit.
\(y=-x^5+1\) which is the same as:
\(y=1-x^5\)

Example 3
Graph \(f(x)=\frac{1}{2}(x-1)^4\)

Solution:
Take it step by step:
Start with the base function of \(y=x^4\)
Replace x by x-1 which will shift it right by 1 unit.
This gives us \(y=(x-1)^4\).
Now, multiple by 1/2 which gives the graph a compression by a factor of 1/2.
We now have \(y=\frac{1}{2}(x-1)^4\)

Finding the Real Zeros of a Polynomial Function
The previous graph and problem shows a polynomial function with four
x-intercepts. Notice that at the x-intercepts the graph must either cross the
x-axis or touch the x-axis. Consequently, between consecutive x-intercepts the
graph is either above the x-axis or below the x-axis. We will make use of this
property of the graph of a polynomial function.

If a polynomial function f is factored completely, it is easy to solve the
equation \(f(x)=0\) using the zero-product property and locate the x-intercepts
of the graph. For example, if \(f(x)=(x-1)^2(x+3)\), then the solution of the
equation becomes:
\(f(x)=(x-1)^2(x+3)=0\) which factored give us: 1 and -3 as intercepts.

Based on this we can make the following observations.
If f is a function and r is a real number for which f(r)-0, then r is called a
real zero of f.

As a consequence, the following statements are equivalent.
1. r is a real zero of a polynomial function f
2. r is an x-intercept of the graph of f
3. x-r is a factor of f

So, the real zeros of a polynomial function are the x-intercepts of its graph
and they are found by solving the equation \(f(x)=0\).

Example 4
Finding a Polynomial from its Zeros
Find a polynomial of degree 3 whose zeros are -3, 2, and 5.

Solution:
If r is a real zero of a polynomial f, then x-r is a factor of f. This means
that x-(-3)=x+3, x-2, and x-5 are factors of f. As a result, any polynomial of
the form \(f(x)=a(x+3)(x-2)(x-5)\) where a is any nonzero real number,
qualifies. The value of a causes a stretch, compression, or reflection, but does
not affect the x-intercepts.

If the same factor x-r occurs more than once, r is called a repeated or multiple
zero of f. More precisely, we have the following definition.
If (x-r)^m is a factor of a polynomial f and (x-r)^{m+1} is not a factor of f,
then r is called a zero of multiplicity m of f.

Example 5
Identifying Zeros and their Multiplicities
For the polynomial \(f(x)=5(x-2)(x+3)^2(x-1/2)^4\)

Solution:
2 is a zero of multiplicity 1 because the exponent on the factor x-2 is 1
-3 is a zero of multiplicity 2 because the exponent on the factor x+3 is 2
1/2 is a zero of multiplicity 4 because the exponent on the factor x-1/2 is 4

Suppose that it is possible to factor completely a polynomial function and
locate all the x-intercepts of its graph(the real zeros of the function). These
x-intercepts then divide the x-axis into open intervals and, on each such
interval, the graph of the polynomial will be either above or below the x-axis.

Example 6
Graphing a Polynomial Using its x-intercepts
For the polynomial \(f(x)=x^2(x-2)\)
1. Find the x and y intercepts of the graph of f
2. Use the x-intercepts to find the intervals on which the graph of f is above
the x-axis and the intervals on which the graph of f is below the x-axis.
3. Locate other points on the graph and connect all the points plotted with a
smooth continuous curve.

Solution:
1. The y-intercept is \(f(0)=0^2(0-2)=0\). The x-intercepts satisfy the
equation \(f(x)=x^2(x-2)=0\) from which we find \(x^2=0\) or \(x-2=0\) and
\(x=0\) or \(x=2\). The x-intercepts are 0 and 2
2. The two x-intercepts divide the x-axis into 3 intervals.
\((-\infty,0)(0,2)(2,\infty)\). Since the graph of f crosses or touches the x-axis
only at x=0 and x=2, it follows that the graph of f is either above the x-axis
f(x)>0 or below the x-axis f(x)<0 on each of these 3 intervals. To see where the
graph lies, we only need to pick a number in each interval, evaluate f there,
and see whether the value is positive(above the x-axis) or negative(below the
x-axis).
3. in constructing a table, we obtained three additional points on the
graph:(-1,-3)(1,-1)(3,9).

Since the graph of \(f(x)=x^2(x-2)\) is below the x-axis on both sides of 0, the
graph of f touches the x-axis at x=0, a zero of multiplicity 2. Since the graph
of f is below the x-axis for x<2 and above the graph for x>2, the graph of f
crosses the x-axis at x=2, a multiplicity of 1.

If r is a zero of even  multiplicity the sign of f(x) does not change from one side to
the other side of r. The graph of f touches the x-axis at r. 

If r is a zero of odd multiplicity, the sign of f(x) changes from one side to
the other side of r. The graph of f crosses the x-axis at r.


Behavior Near a Zero
We have just learned how the multiplicity of a zero can be used to determine
whether the graph of a function touches or crosses the x-axis at the zero.
However, we can learn more about the behavior of the graph near its zeros than
just whether the graph crosses or touches the x-axis. 

Consider the function\(f(x)=x^2(x-2)\). The zeros of f are 0 and 2. We can see
that the points on the graph of \(f(x)=x^2(x-2)\) and the points on the graph of
\(y=-2x^2\) are indistinguishable near x=0. So, \(y=-2x^2\) describes the
behavior of the graph of \(f(x)=x^2(x-2)\) near x=0.

But how did we know that the function \(f(x)=x^2(x-2)\) behaves like \(y=-2x^2\)
when x is close to 0? Where did \(y=-2x^2\) comes form? Because the zero, 0,
comes from the factor \(x^2\), we evaluate all factors in the function f at 0
with the exception of \(x^2\).
\(f(x)=x^2(x-2)\) 
The factor \(x^2\) gives rise to the zero, so we keep the factor \(x^2\) and let
x=0 in the remaining factors to find the behavior near 0.
\(f(x)=x^2(0-2) = -2x^2\)

This tells us that the graph of \(f(x)=x^2(x-2)\) will behave like the graph of
\(y=-2x^2\) near x=0. Now let us discuss the behavior of \(f(x)=x^2(x-2)\) near
x=2, the other zero. Because the zero, 2, comes from the factor x-2, we
evaluate all factors of the function f at 2, with the exception of x-2.
\(f(x)=x^2(x-2) = 2^2(x-2) = 4(x-2)\)
The factor x-2 gives rise to the zero, so we keep the factor x-2 and let z=2 in
the remaining factors to find the behavior near 2.

So the graph of \(f(x)=x^2(x-2)\) will behave like the graph of \(y=4(x-2)\)
near x=2. We can see that the points on the graph of \(f(x)=x^2(x-2)\) and the
points on the graph of \(y=4(x-2)\) are indistinguishable near x=2. So
\(y=4(x-2)\), a line with slope 4, describes the behavior of the graph of
\(f(x)=x^2(x-2)\) near x=2.

By determining the multiplicity of a real zero, we determine that the graph
crosses or touches the x-axis at the zero. By determining the behavior of the
graph near the real zero, we determine how the graph touches or crosses the
x-axis.

Turning Points
When looking at the previous graph, we cannot be sure how low the graph actually
goes between x=0 and x=2. but we do know that somewhere in the interval (0,2)
the graph of f must change direction (from decreasing to increasing). The points
at which a graph changes direction are called turning points. In calculus, such
points are called local maxima and local minima, and techniques for locating
them are given. So we shall not ask for the location of turning points in our
graphs. Instead, we will use the following result from calculus, which tells us
the maximum number of turning points that the graph of a polynomial function can
have.

If f is a polynomial function of degree n, then f has at most n-1 turning
points.
If the graph of a polynomial function f has n-1 turning points, the degree of f
is at least n.

For example, the graph of \(f(x)=x^2(x-2)\) is the graph of a polynomial of degree
3 and 3-1=2 turning points: one at (0,0) and the other somewhere between x=0 and
x=2.

Example 7
Identifying the graph of a polynomial function
Which of the graphs could be the graph of a polynomial function? For those that
could, list the real zeros and state the least degree the polynomial can have.
For those that could not, say why not.

Solution:
1. This cannot be the graph of a polynomial function because of the gap that
occurs at x=-1. Remember, the graph of a polynomial function is continuous with
no gaps or holes.
2. This could be the graph of a polynomial function. It has 3 real zeros at -2,
1, and 2. Since the graph has 2 turning points, the degree of the polynomial
function must be at least 3.
3. This cannot be the graph of a polynomial function because of the cusp at x=1.
Remember, the graph of a polynomial function is smooth.
4. This could be the graph of a polynomial function. It has 2 real zeros at -2
and 1. Since the graph has 3 turning points, the degree of the polynomial
function is at least 4.

For very large values of x, either positive or negative, the graph of
\(f(x)=x^2(x-2)\) looks like the graph of \(y-=x^3\). To see why, we write f in
the form:
\(f(x)=x^2(x-2)=x^3-2x^2=x^3(1-\frac{2}{x})\)
Now, for large values of x, either positive or negative, the term 2/x is close to
0, so for large values of x:
\(f(x)=x^3-2x^2=x^3(1-\frac{2}{x})=x^3\)
The behavior of the graph of a function for large values of x, either positive
or negative, is referred to as its end behavior.

Example 8
Identifying the graph of a polynomial function
Which of the graphs could be the graph of:
\(f(x)=x^4+5x^3+5x^2-5x-6\)

Solution:
The y-intercept of f is f(0)=-6 so we can eliminate any graph whose y-intercept
is positive. Being positive means the curve crosses the y-axis above 0.
We don't have any methods for finding the x-intercept of f, so we move on to
investigate the turning points of each graph. since f is of degree 4, the graph
of f has at most 3 turning points. So, we eliminate the graph that has 5 turning
points. 
Now, we look at end behavior. For large values of x, the graph of f will behave
like the graph of \(y=x^4\). This eliminates the graph whose end behavior is
like the graph of \(y=-x^4\). 
Only the graph left is the graph which has 3 turning points and behaves like
\(x^4\).

Summary of Polynomial Functions
Degree of polynomial f: n
Maximum number of turning points: n-1
At a zero of even multiplicity: the graph of f touches the x axis
At a zero of odd multiplicity: the graph of f crosses the x-axis.
Between zeros, the graph of f is either above or below the x-axis
End behavior: For large |x|, the graph of f behaves like the graph of of
\(y=a_{n}x^{n}\).

Example 9
Analyzing the Graph of a Polynomial Function
For the polynomial \(f(x)=x^3+x^2-12x\) :
1. Find the x and y-intercepts of the graph of f
2. Determine whether the graph crosses or touches the x-axis at each
   x-intercept
3. End behavior: Find the power function that the graph of f resembles for large
   values of |x|
4. Determine the maximum number of turning points on the graph of f
5. Determine the behavior of the graph of f near each x-intercept
6. Put all the information together to obtain the graph of f

Solution:
1. The y-intercept is 0 because \(f(0)=0^3 +0^2-12(0)\)
   To find the x-intercepts, if any, we first factor f
   \(x(x^2+x-12 = x(x+4)(x-3)\) so x=0, -4, and 3
2. Since each real zero is of multiplicity 1, the graph of f will cross the
   x-axis at each x-intercept
3. End behavior: the graph of f resembles that of the power function \(y=x^3\)
   for large values of |x|
4. The graph of f will contain at most two turning points because \(x^3\) is the
   highest exponent
5. The three x-intercepts are -4, 0, and 3
   Near -4: \(-4(x+4)(-4-3) = 28(x+4)\) A line with slope 28
   Near 0: \(x(0+4)(0-3)=-12x\) A line with slope -12
   Near 3: \(3(3+4)(x-3)=21(x-3)\) A line with slope 21
6. Put all of this together on a piece of paper then use a calculator to graph
   and see if it is close.

Example 10
Analyzing the Graph of a Polynomial Function
\(f(x)=x^2(x-4)(x+1)\)

Solution:
1. The y-intercept is 0 because f(0)=0
2. The x-intercepts are 0, 4, and -1
   The intercept 0 is a multiplicity of 2 so the graph of f will touch the
   x-axis 4 and -1 are zeros of multiplicity 1, so the graph of f will cross    the x-axis at 4 and -1.
3. End behavior: the graph of f resembles that of the power function \(y=x^4\)
   for large values of |x| 
4. The graph of f will contain at most three turning points because of \(x^4\)
5. The three x-intercepts are -1, 0, and 4
   Near -1: \(-1^2(-1-4)(x+1)=-5(x+1)\) A line with slope -5
   Near 0: \(x^2(0-4)(0+1)=-4x^2\) A parabola opening down
   Near 4: \(4^2(x-4)(4+1)=80(x-4)\) A line with slope 80
6. Graph it on paper then with a calculator

Summary for Analyzing the Graph of a Polynomial
1. Find the y-intercept by letting x=0 and finding the value of f(0)
   Find the x-intercepts, if any, by solving the equation f(x)=0
2. Determine whether the graph of f crosses or touches the x-axis at each
   x-intercept
3. End behavior: find the power function that the graph of f resembles for large
   values of |x|
4. Determine the maximum number of turning points on the graph of f
5. Determine the behavior of the graph of f near each x-intercept
6. Put all the information together to obtain the graph of f

For polynomial functions that have non-integer coefficients and for polynomials
that are not easily factored, we utilize a graphing utility early in the
analysis of the graph. This is because the amount of information that can be
obtained from algebraic analysis is limited.

Properties of Rational Functions

Ratios of integers are called rational numbers. Ratios of polynomial functions
are called rational functions. A rational function is of the form
\(r(x)=\frac{p(x)}{q(x)}\). P and Q are polynomial functions and q is not the
zero polynomial. The domain of a rational function is the set of all real
numbers except those for which the denominator is zero.

Example 1
Find the domain of a rational function
1. The domain of \(r(x)=\frac{2x^2-4}{x+5}\) is the set of all real numbers
except -5. The domain is \({x|x\not=-5}\).
2. The domain of \(r(x)=\frac{1}{x^2-4}\) is the set of all real numbers except
-2 so the domain is \({x|x\not=-2,x\not=2}\).
3. The domain of \(r(x)=\frac{x^3}{x^2+1}\) is the set of all real numbers.
4. The domain of \(r(x)=\frac{3}{x^2+2}\) is the set of all real numbers.
5. The domain of \(r(x)=\frac{x^2-1}{x-1}\) is the set of all real numbers
except 1 so the domain is \({x|x\not=1}\).

If \(r(x)=\frac{p(x)}{q(x)}\) is a rational function and if p and q have no
common factors, then the rational function is said to be in lowest terms. For a
rational function in lowest terms, the real zeros, if any, of the numerator are
the x-intercepts of the graph will play a major role in the graph. The real
zeros of the denominator also play a major role in the graph. 

Example 2
Graphing \(y=\frac{1}{x^2}\)

Solution:
The domain is the set of all real numbers except 0. The graph has no y-intercept
because x can never equal 0. The graph has no x-intercept because the equation
\(y=0\) has no solution. Therefore, the graph will not cross either of the
coordinate axes. Because:
\(h(-x) = \frac{1}{(-x)^2} = \frac{1}{x^2} = h(x)\)
It is an even function, so its graph is symmetric with respect to the y-axis. 

Example 3
Using transformations to graph a rational function
Graph the rational function \(r(x) = \frac{1}{(x-2)^2 + 1}\)

Solution:
First, we take note of the fact that the domain of r is the set of all real
numbers except x=2. To graph r, we start with the graph of \(y=\frac{1}{x^2}\).
Use a calculator to graph the function.

Asymptotes
In the previous graph, notice that as the values of x become more negative as x
approaches negative infinity, the values approach 1. In fact, we can conclude
the following:
1. As x approaches negative infinity, the values of r(x) approach 1
2. As x approaches 2, the values of r(x) approach infinity
3. As x approaches infinity, the values of r(x) approach 1
This behavior of the graph is shown by the vertical line x=2 and the horizontal
line y=1. These lines are called asymptotes of the graph.

A horizontal asymptote, when it occurs, describes the end behavior of the graph
as x approaches infinity or as x approaches negative infinity. The graph of a
function may intersect a horizontal asymptote. A vertical asymptote, when it
occurs, describes the behavior of the graph when x is close to some number. The
graph of a function will never intersect a vertical asymptote.

There is a third possibility. If the value of a rational function approaches a
linear expression, then the line y=ax+b is an oblique asymptote. An oblique
asymptote, when it occurs, describes the end behavior of the graph. The graph of
a function may intersect an oblique asymptote.

Find the Vertical Asymptotes of a Rational Function
The vertical asymptotes of a rational function, in lowest terms, are located at
the real zeros of the denominator. Suppose that r is a real zero of q, so x-r is
a factor of q. As x approaches r, the values of x-r approach 0, causing the
ratio to become unbounded, or head towards infinity. Based on the definition, we
conclude that the line x=r is a vertical asymptote. 

Example 4
Finding vertical asymptotes
Find the vertical asymptotes of the graph of each rational function
1. \(r(x)=\frac{x}{x^2-4}\)
2. \(r(x)=\frac{x+3}{x-1}\)
3. \(r(x)=\frac{x^2}{x^2+1}\)
4. \(r(x)=\frac{x^2-9}{x^2+4x-21}\)

Solution:
1. The function is in lowest terms and the zeros of the denominator are -2 and
   The lines x=-2 and x=2 are the vertical asymptotes of the graph.
2. The function is in lowest terms and the only zero of the denominator is 1.
   The line x=1 is the vertical asymptote of the graph.
3. The function is in lowest terms and the denominator has no real zeros,
   because the equation has no real solutions. The graph has no vertical      asymptotes
4. Factor to get it to lowest terms. 
   \(r(x)=\frac{x^2-9}{x^2+4x-21} = \frac{(x+3)(x-3)}{(x+7)(x-3)} =
   \frac{x+3}{x+7}\) The only zero of the denominator in lowest terms is -7. The
   line x=-7 is the only vertical asymptote of the graph.

Rational functions can have no vertical asymptotes, one vertical asymptote, or
multiple vertical asymptotes. However, the graph of a rational function will
never intersect any of its vertical asymptotes. 

Find the Horizontal or Oblique Asymptotes of a Rational Function
The procedure for finding horizontal and oblique asymptotes is somewhat more
involved. To find such asymptotes, we need to know how the values of a function
behave as x a[[roaches negative infinity or as x approaches infinity. 

If a rational function is proper, that is, if the degree of the numerator is
less than the degree of the denominator, then as x approaches negative infinity
or as x approaches infinity the value of the function approaches 0.
Consequently, the line y=0(x-axis) is a horizontal asymptote of the graph. If a
rational function is proper, the line y=0 is a horizontal asymptote of its graph.

Example 5
Finding Horizontal Asymptotes
Find the horizontal asymptotes, if any, of the graph of:
\(f(x)=\frac{x-12}{4x^2+x+1}\)

Solution:
Since the degree of the numerator is 1, it is less than the degree of the
denominator which is 2. So this is a proper function. That means that the line
y=0 is a horizontal asymptote of the graph.

To see why y=0 is a horizontal asymptote of the function, we need to investigate
the behavior as x approaches negative infinity and as x approaches infinity.
When |x| is unbounded, the numerator, which is x-12, can be approximated by the
power function y=x, while the denominator, which is \(4x^2+x+1\), can be
approximated by the power function \(y=4x^2\). We find:
\(f(x)=\frac{x-12}{4x^2+x+1} = \frac{x}{4x^2} = \frac{1}{4x}\) as it approaches
0. This shows that the line y=0 is a horizontal asymptote of the graph.

If a rational function is improper, or if the degree of the numerator is greater
than the degree of the denominator, we must use long division to write the
rational function as the sum of a polynomial plus a proper rational function.
1. If f(x)=b, a constant, the line y=b is a horizontal asymptote of the graph
2. If f(x)=ax+b, the line y=ax+b is an oblique asymptote of the graph
3. In all other cases, the graph of r approaches the graph of f, and there are
   no horizontal or oblique asymptotes.

Example 6
Finding Horizontal or Oblique Asymptotes
Find the horizontal or oblique asymptote of the graph of:
\(f(x)=\frac{3x^4-x^2}{x^3-x^2+1}\)

Solution:
Since the degree of the numerator, 4, is larger than the degree of the
denominator, 3, the rational function is improper. To find any horizontal or or
oblique asymptotes, use long division to get 3x+3, which is the oblique
asymptote.

Example 7
Finding Horizontal or Oblique Asymptotes
Find asymptotes of:
\(f(x)=\frac{8x^2-x+2}{4x^2-1}\)

Solution:
Since the degree of the numerator, 2, equals the degree of the denominator,2,
the rational function is improper. To find any horizontal or oblique asymptotes,
use long division. We get y=2 as a horizontal asymptote of the graph.

We note that the quotient 2 obtained by long division is the quotient of the
leading coefficients of the numerator polynomial and the denominator polynomial.
This means that we can avoid the long division process for rational functions
whose numerator and denominator are of the same degree and conclude that the
quotient of the leading coefficients will give us the horizontal asymptote.

Example 8
Finding the Horizontal or Oblique Asymptotes
\(f(x)=\frac{2x^5-x+2}{x^3-1}\)

Solution:
Since the degree of the denominator, 5, is larger than the degree of the
denominator, 3, the rational function is improper. To find any horizontal or
oblique asymptotes, use long division. \(2x^2-1\). Since this is not a linear
function, the function has no horizontal or oblique asymptotes.

1. If the degree of the numerator is less than the degree of the denominator, it
is a proper rational function, and the graph will have the horizontal asymptote
y=0.
2. If the degree of the numerator is greater than or equal to the degree of the
denominator, then it is an improper function. Use long division.
 A. if the degree of the numerator equals the degree of the denominator, the
 quotient obtained will be the number\(\frac{a_{n}}{b_{m}}\) and the line
 \(y=\frac{a_{n}}{b_{m}}\) is a horizontal asymptote.
 B. If the degree of the numerator is more than the degree of the
 denominator, the quotient obtained is of the form ax+b and the line y=ax+b
 is an oblique asymptote.
 C. If the degree of the numerator is two or more than the degree of the
 denominator, the quotient obtained is a polynomial of degree 2 or higher,
 and the function has neither a horizontal nor an oblique asymptote. In this
 case, for |x| unbounded, the graph will behave like the graph of the
 quotient.

Graphs of Rational Functions

Steps for Analyzing a Graph
1. Factor the numerator and denominator and find the domain. If 0 is in the
domain, find the y-intercept, f(0), and plot it.

2. Write the function in lowest terms as \(\frac{p(x)}{q(x)}\) and find the real
zeros of the numerator. That is finding the real solution of the equation
p(x)=0. These are the x-intercepts of the graph. Determine the behavior of the
graph near each x-intercept, using the same procedure as for polynomial
functions. Plot each x-intercept and indicate the behavior of the graph.

3. With the function written in lowest terms, find the real zeros of the
equation q(x)=0. These determine the vertical asymptotes of the graph. Graph
each vertical asymptote using a dashed line.

4. Locate any horizontal or oblique asymptotes using the procedure given in the
previous section. Graph the asymptotes using a dashed line. Determine the points
at which the graph intersects these asymptotes. Plot any such points.

5. Using the real zeros of the numerator and the denominator of the given
equation, divide the x-axis into intervals and determine where the graph is
above the x-axis and where it is below the x-axis by choosing a number in each
interval and evaluating the function there. Plot the points found.

6. Analyze the behavior of the graph near each asymptote and indicate this
behavior in the graph.

7. Put all the information together to obtain the graph of the function given.

Example 1
Analyze the graph of a rational function
\(r(x)=\frac{x-1}{x^2-4}\)

Solution:
1. We factor the numerator and denominator:
\(r(x)=\frac{x-1}{(x+2)(x-2)}\)
The domain is \({x|x \not=2, x  \not =-2}\)
The y-intercept is \(r(0)=\frac{-1}{-4} = \frac{1}{4}\)
Plot the point \(0,\frac{1}{4}\)

2. Now that r is in lowest terms, the real zeros of the numerator satisfies the
equation x-1=0. The only x-intercept is 1.
Near 1: \(r(x)=\frac{x-1}{(x+2)(x-2)} = \frac{x-1}{(1+2)(1-2} =
\frac-{1}{3}(x-1)\)
Plot the point (1,0) and indicate a line with slope \(-\frac{1}{3}\) there.

3. R is in lowest terms. The real zeros of the denominator are the real
solutions of the equation (x+2)(x-2)=0, so -2 and 2. The graph of r has two
vertical asymptotes, the lines x=2 and x=-2. Graph each of these asymptotes
using dashed lines.

4. The degree of the numerator is less than the degree of the denominator, so r
is proper and the line y=0 (the x axis) is a horizontal asymptote of the graph.
Indicate this line by graphing y=0 using a dashed line. To determine if the
graph of r intersects the horizontal asymptote, we solve the equation r(x)=0.
\(\frac{x-1}{x^2-4}=0\)
\(x-1=0 >> x=1\)
The only solution is x=1. so the graph of r intersects the horizontal asymptote
at (1,0). We have already plotted this point.

5. The zero of the denominator, 1, and the zeros of the denominator, -2 and 2,
divide the x-axis into 4 intervals. Pick a point from each interval and solve
r(x). If the solution is negative then it is below the x-axis and if it is
positive then it is above the x-axis. 

6. Next, we determine the behavior of the graph near the asymptotes.
   A. Since the x-axis is a horizontal asymptote and the graph lies below the
   x-axis for x<-2, we can sketch a portion of the graph by placing a small
   arrow to the far left and under the x-axis.
   B. Since the line x=-2 is a vertical asymptote and the graph lies below the
   x-axis for x<-2, we continue by placing an arrow well below the x-axis and
   approaching the line x=-2 on the left.
   C. Since the graph is above the x-axis for -2<x<1 and x=-2 is a vertical
   asymptote, the graph will continue on the right of x=-2 at the top.

Example 2
Analyze the graph of a rational function
\(r(x)=\frac{x^2-1}{x}\)

Solution:
1. The domain of r is \({x|x\not=0}\). Because x cannot equal 0, there is not
y-intercept. Now factor r to obtain \(r(x)=\frac{(x+1)(x-1)}{x}\)

2. R is in lowest terms. Solving the equation r(x)=0, we find the graph has two
x-intercepts, -1 and 1.
Near -1: \(r(x)=\frac{(x+1)(x-1)}{x} = \frac{(x+1)(-1-1)}{-1} = 2(x+1)\)
Near 1: \(r(x)=\frac{(x+1)(x-1)}{x} = \frac{(1+1)(x-1)}{1} = 2(x-1)\)
Plot the point (-1,0) and indicate a line with slope 2 there. Plot the point
(1,0) and indicate a line with slope 2 there.

3. R is in lowest terms, so the graph of r has the line x=0 (the y-axis) as a
vertical asymptote. Graph x=0 using a dashed line.

4. The rational function r is improper, since the degree of the numerator, 2, is
larger than the degree of the denominator, 1. To find any horizontal or oblique
asymptotes, use long division. There is no solution so the graph of r does not
intersect the line y=x.

5. The zeros of the numerator are -1 and 1. The zero of the denominator is 0. We
use these values to divide the x-axis into four intervals. Pick a value in each
interval and solve for r(x). If the value is negative then the point is below
the x-axis. If the point is positive then the point is above the x-axis. 

6. Since the graph of r is below the x-axis for x<-1 and is above the x-axis for
x>1, the graph of r will approach the line y=x. Since the graph of r is above
the x-axis for -1<x<0, the graph of r will approach the vertical asymptote x=0
at the top to the left of x=0. Since the graph of r is below the x-axis for
0<x<1, the graph of r will approach the vertical asymptote x=0 at the bottom
right of x=0.

Example 3
Analyze the graph of a rational function
\(r(x)=\frac{x^4+1}{x^2}\)

Solution:
1. R is completely factored. The domain of r is \({x|x\not=0}\). There is no
y-intercept.

2. R is in lowest terms. Since \(x^4+1=0\) has no real solutions, there are no
x-intercepts.

3. R is in lowest terms, so x=0 (the y-axis) is a vertical asymptote of r. Graph
the line x=0 using dashes.

4. The rational function r is improper. To find any horizontal or oblique
asymptotes, use long division. We find the quotient to be \(x^2\), so the graph
has no horizontal or oblique asymptotes. However, the graph of r will approach
the graph of \(y=x^2\) as x approaches negative infinity and as x approaches
infinity. The graph of r does not intersect \(y=x^2\). Graph \(y=x^2\) using
dashes.

5. The numerator has no zeros, and the denominator has one zero at 0. We divide
the x-axis into the two intervals. Plot the points (-1,2) and (1,2).

6. Since the graph of r is above the x-axis and does not intersect \(y=x^2\), we
place arrows above \(y=x^2\). Also, since the graph of r is above the x-axis, it
will approach the vertical asymptote x=0 at the top to the left of x=0 and at
the top to the right of x=0.

Example 4
Analyze the graph of a rational function
\(r(x)=\frac{3x^2-3x}{x^2+x-12}\)

Solution:
1. We factor r to get \(r(x)=\frac{3x(x-1)}{(x+4)(x-3)}\). The domain of r is
\({x|x\not=-4,x\not=3}\). The y-intercept is r(0)=0. Plot the point (0,0).

2. R is in lowest terms. Since the real solutions of the equation 3x(x-1)=0 are
x=0 and x=1, the graph has two x-intercepts, 0 and 1. We determine the behavior
of the graph of r near each x-intercept.
Near0:\(r(x)=\frac{3x(x-1)}{(x+4)(x-3)}=\frac{3x(0-1)}{(0+4)(0-3)}=\frac{1}{4}x\)
Near1:\(r(x)=\frac{3x(x-1)}{(x+4)(x-3)}=\frac{3(1)(x-1)}{(1+4)(1-3)}=-\frac{3}{10}(x-1)\)
Plot the point (0,0) and show a line with slope \(\frac{1}{4}\) there. Plot the
point (1,0) and show a line with slope \(-\frac{3}{10}\) there.

3. R is in lowest terms. Since the real solutions of the equation (x+4)(x-3)=0
are x=-4 and x=3, the graph of r has two vertical asymptotes, the lines x=-4 and
x=3. Plot these lines using dashes.

4. Since the degree of the numerator equals the degree of the denominator, the
graph has a horizontal asymptote. To find it, we either use long division or
form the quotient of the leading coefficient of the numerator, 3, and the
leading coefficient of the denominator, 1. The graph of r has the horizontal
asymptote y=3. To find out whether the graph of r intersects the asymptote, we
solve the equation r(x)=3.
\(r(x)=\frac{3x^2-3x}{x^2+x-12}=3\)
\(3x^2-3x=3x^2+3x-36\)
\(-6x=-36\)
\(x=6\)
The graph intersects the line y=3 only at x=6, and (6,3) is a point on the graph
of r. Plot the point (6,3) and the line y=3 using dashes.

5. The zeros of the numerator, 0 and 1, and the zeros of the denominator, -4 and
3, divide the x-axis into 5 intervals. Choose a number in each interval, solve
r(x) for each number chosen. If solution is positive then the point is above the
x-axis but if the solution is negative then the point is below the x-axis. 

6. Since the graph of r is above the x-axis for x<-4 and only crosses the line
y=3 at (6,3), as x approaches negative infinity the graph of r will approach the
horizontal asymptote y=3 from above. The graph of r will approach the vertical
asymptote x=-4 at the top to the left of x=-4 and at the bottom to the right of
x=-4. The graph of r will approach the vertical asymptote x=3 at the bottom to
the left of x=3 and at the top to the right of x=3. We do not know whether the
graph of r crosses or touches the line y=3 at (6,3). To see whether the graph
crosses or touches the line y=3, we plot an additional point to the right of
(6,3) at x=6. Because (6,3) is the only point where the graph of r intersects
the asymptote y=3, the graph must approach the line y=3 from below as x
approaches infinity. 

Example 5
Analyze the graph of a rational function with a hole

Solution:
1. We factor r and obtain: \(r(x)=\frac{(2x-1)(x-2)}{x^2-4}\)
The domain of r is \({x|x\not=-2,x\not=2}\). The y-intercept is r(0)=-1/2. Plot
the point (0,-1/2).

2. In lowest terms, \(r(x)=\frac{2x-1}{x+2}\) \(x\not=-2\)
The graph has one x-intercept: 1/2.
Near 1/2: \(r(x)=\frac{2x-1}{x+2} = \frac{2x-1}{\frac{1}{2}+2} = 2/5(2x-1)\)
Plot the point (1/2,0) showing a line with slope 4/5.

3. Look at r in lowest terms. The graph has one vertical asymptote, x=-2, since
x+2 is the only factor of the denominator of r(x) in lowest terms. Remember,
though, the rational function is undefined at both x=2 and x=-2. Graph the line
x=-2 using dashes.

4. Since the degree of the numerator equals the degree of the denominator, the
graph has a horizontal asymptote. To find it, we use long division or form the
quotient of the leading coefficient of the numerator 2, and the leading
coefficient of the denominator, 1. The graph of r has the horizontal asymptote
y=2. Graph the line y=2 using dashes.
\(r(x)=\frac{2x-1}{x+2} = 2\)
\(2x-1=2(x+2)\)
\(2x-1=2x+4\)
\(-1=4\) Not a solution. The graph doe snot intersect the line y=2.

5. Look at the given expression for r. The zeros of the numerator and
denominator, -2,1/2, and 2, divide the x-axis into four intervals. Plot these
points. 

6. We know the graph of r is above the x-axis for x<-2. We know the graph of r
does not intersect the asymptote y=2. Therefore, the graph of r will approach
y=2 from above as x approaches negative infinity and will approach the vertical
asymptote x=-2 at the top from the left. Since the graph of r is below the
x-axis for -2<x<1/2, the graph of r will approach x=-2 at the bottom from the
right. Finally, since the graph of r is above the x-axis for x>1/2 and does not
intersect the horizontal asymptote y=2, the graph of r will approach y=2 from below as x approaches infinity.

The real zeros of the denominator of a rational function give rise to either
vertical asymptotes or holes on the graph. 

Example 6
Constructing a rational function from its graph

Solution:
The numerator of a rational function in lowest terms determines the x-intercepts
of its graph. This graph has x-intercepts -2(even multiplicity, graph touches
the x-axis) and 5(odd multiplicity, graph crosses the x-axis). So one
possibility for the numerator is \(p(x)=(x+2)^2(x-5)\). 

The denominator of a rational function in lowest terms determines the vertical
asymptotes of its graph. The vertical asymptotes of the graph are x=-5 and x=2.
Since r(x) approaches infinity to the left of x=-5 and r(x) approaches negative
infinity to the right of x=-5, then x+5 is a factor of odd multiplicity in q(x).
Also, because r(x) approaches negative infinity on both sides of x=2, then x-2
is a factor of even multiplicity in q(x). A possibility for the denominator is
\(q(x)=(x+5)(x-2)^2\). So far, we have:
\(r(x)=\frac{(x+2)^2(x-5)}{(x+5)(x-2)^2}\).

The horizontal asymptote of the graph is y=2, so we know that the degree of the
numerator must equal the degree of the denominator and the quotient of leading
coefficients must 2/1. This leads to:
\(r(x)=\frac{2(x+2)^2(x-5)}{(x+5)(x-2)^2}\)

Polynomial Inequalities

We solve inequalities that involve polynomials of degree 3 and higher, as well
as some that involve rational expressions. To solve such inequalities, we use
the information obtained previously about the graph of polynomial and rational
functions. 

Suppose that the polynomial or rational inequality is of the form:
\(f(x)<0,f(x)>0,f(x)\leq0,f(x)\geq0\)
Locate the zeros of f if f is a polynomial function, and locate the zeros of the
numerator and the denominator if f is a rational function. If we use these zeros
to divide the real number line into intervals, we know that on each interval the
graph of f is either above the x-axis or below the x-axis. In other words, we
have found the solution of the inequality.

Steps for Solving Polynomial Inequalities
1. Write the inequality so that a polynomial or rational expression is on the
left side and zero is on the right side. For rational expressions, be sure that
the left side is written as a single quotient and find its domain.

2. Determine the real numbers at which the expression on the left side equals
zero and, if the expression is rational, the real numbers at which the
expression on the left side is undefined.

3. Use the numbers found in step 2 to separate the real number line into
intervals.

4. Select a number in each interval and evaluate the function at the number.
   A. If the value of f is positive, then f(x)>0 for all numbers in the interval
   B. If the value of f is negative then f(x)<0 for all numbers in the interval
If the inequality is not strict, include the solutions of f(x)=0 in the solution
set.

Example 1
Solving a Polynomial Inequality
\(x^4 \leq 4x^2\)

Solution:
Rearrange the inequality so that 0 is on the right side.
\(x^4 \leq 4x^2\)
\(x^4-4x^2 \leq 0\)
This inequality is equivalent to the one that we wish to solve.
Find the real zeros of \(f(x)=x^4-4x^2\) by solving its equation.
\(x^4-4x^2=0\)
\(x^2(x^2-4)=0\)
\(x^2(x+2)(x-2)=0\)
x=0 or x=-2 or x=2
We use these zeros to separate the real number line into four intervals.
select a number in each interval and evaluate \(f(x)=x^4-4x^2\) to determine if
f(x) is positive or negative. 
We know that f(x)<0 for x in the intervals (-2,0) and (2,0), for all x such that
-2<x<0 or 0<x<2. however, because the original inequality is not strict, numbers
that satisfy the equation are also solution of the inequality. So, we include 0,
-2, and 2. The solution set of the given inequality is \({x|x-2 \leq x \leq
2}\). 

Example 2
Solving a polynomial inequality
\(x^4 > x\)

Solution:
Rearrange the inequality so that 0 is on the right side.
\(x^4-x > 0\)
This inequality is equivalent to the one that we wish to solve.
Find the real zeros of \(f(x)=x^4-x\) by solving \(x^4-x=0\)
\(x^4-x=0 \longrightarrow x(x^3-1)=0 \longrightarrow x(x-1)(x^2+x+1)=0\)
x=0 or x=1 The squared term is a quadratic and of course does not have any real
solutions.
We use the zeros 0 and 1 to separate the real number line into three intervals.
We choose a number in each interval and evaluate \(f(x)=x^4-x\) to determine if
the solution is positive or negative.
We know that f(x)>0 for all x in negative infinity to 0 and from 1 to infinity.
Because the original inequality is strict, the solution set of the given
inequality is \({x|x,0,x>1}\).

Example 3
Solving a rational inequality
Solve \(\frac{(x+3)(2-x)}{(x-1)^2}>0\) and graph the solution set.

Solution:
The domain of the variable x is \({x|x\not=1}\). The inequality is already in a
form with 0 on the right side.
Let \(f(x)=\frac{(x+3)(2-x)}{(x-1)^2}\). The real zeros of the numerator of f
are -3 and 2. The real zero of the denominator is 1. 
We use the zeros -3, 1 and 2 to separate the real number line into four
intervals.
Select a number in each interval and evaluate the number in
\(f(x)=\frac{(x+3)(2-x)}{(x-1)^2}\) to determine in f(x) is positive or
negative.
We know that f(x)>0 for all x in (-3,1) to (1,2). Because the original
inequality is strict, the solution set of the given inequality is
\({x|-3<x<2,x\not=1}\). Notice the hole at x=1 to indicate that 1 is to be
excluded.

Example 4
Solving a rational inequality
\(\frac{4x+5}{x+2} \geq 3\)

Solution:
The domain of the variable x is \({x|x\not=-2}\). Rearrange the inequality so
that 0 is on the right side. Then express the left side as a single quotient.
\(\frac{4x+5}{x+2}-3 \geq 0\)
Multiply 3 by \(\frac{x+2}{x+2}\)
\(\frac{4x+5}{x+2}-3(\frac{x+2}{x+2} \geq 0 \)
Write as a single quotient
\(\frac{4x+5-3x+6}{x+2} \geq 0\)
Combine like terms
\(\frac{x-1}{x+2} \geq 0\)
The domain of the variable is \({x|x\not=-2}\).
Let \(f(x)=\frac{x-1}{x+2}\). The zero of the numerator is 1. The zero of the
denominator is -2.
We use the zeros -2, and 1 to separate the real number line into three
intervals.
Select a number in each interval and evaluate \(f(x)=\frac{4x+5}{x+2}-3\) to
determine if f(x) is positive or negative.
We know that f(x)>0 for all x from negative \((-\infty,-2)\) and \(1,\infty)\).
Because the original inequality is not strict, numbers that satisfy the equation
\(f(x)=\frac{x-1}{x+2}=0\) are also solutions of the inequality. Since
\(\frac{x-1}{x+2}=0\) only if x=1, we conclude that the solution set is
\({x|x<-2,x\geq1}\).

Real Zeros of Polynomials

In this section, we discuss the techniques that can be used to find the real
zeros of a polynomial function. Recall that if r is a real zero of a polynomial
function f then f(r)=0, r is an x-intercept of the graph of f, and r is a
solution of the equation f(x)=0.

For polynomial and rational functions, we have seen the importance of the real
zeros for graphing. In most cases, the real zeros of a polynomial function are
difficult to find using algebraic methods. No nice formulas like the quadratic
formula are available to help us find zeros for polynomials of degree 3 or
higher. Formulas do exist for solving any third or fourth degree polynomial
equation, but they are complicated to use. No general formulas exist for
polynomial equations of degree 5 or higher.

Remainder and Factor Theorems
When we divide one polynomial by another, we obtain a quotient polynomial and a
remainder, the remainder being either the zero polynomial or a polynomial whose
degree is less than the degree of the divisor. To check our work, we do:
(quotient)(divisor) + remainder = dividend
This checking routine is the basis for a famous theorem called the division
algorithm for polynomials.

Remainder Theorem
Let f be a polynomial function. If f(x) is divided by x-c, then the remainder is
f(c).

Example 1
Using the remainder theorem
Find the remainder if \(f(x)=x^3-4x^2-5\) when divided by x-3 and x+2

Solution:
We could use long division, but who wants to do that, it is easier to use the
remainder theorem.
\(f(3)=(3)^3-4(3)^2-5 = 27-36-5 = 14\)
The remainder is 14

To find the remainder when f(x) is divided by x+2 = x-(-2) = -2
\(f(-2) = (-2)^3-4(-2)^2-5 = -8-16-5 = -29\)
The remainder is -29

Factor Theorem
Let f be a polynomial function. Then x-c is a factor of f(x) if ands only if
f(c)=0. If f(c)=0, then x-c is a factor of f(x). If x-c is a factor of f(x),
then f(c)=0.

Suppose that f(c)=0. Then, we have \(f(x)=(x-c)q(x)\) for some polynomial q(x).
That is, x-c is a factor of f(x). Suppose that x-c is a factopr of f(x). Then
there is a polynomial function q such that \(f(x)=(x-c)q)x)\). Replacing x by c,
we find that \(f(c)=(c-c)q(c) = 0*q(c) = 0\). That completes the proof. One use
of the factor theorem is to determine whether a polynomial has a particular
factor.

Example 2
Use the factor theorem to determine whether a function has a factor
\(f(x)=2x^3-x^2+2x-3\)
For x-1 and x+3

Solution:
That factor theorem states that if f(c)=0 then x-c is a factor
Because x-1 is of the form x-c with c=1, we find the value of f(1). We choose to
use substitution.
\(f(1) = 2(1)^3-(1)^2+2(1)-3 = 2-1+2-3 = 0\)
By the factor theorem, x-1 is a factor of x.

To test the factor x+3, we first need to write it in the form of x-c. Since
x+3=x-(-3), we find the value of f(-3). Because f(-3) =-72 and not 0, we
conclude from the factor theorem that x-(-3) or x+3 is not a factor of f(x).

Number of Real Zeros
The next theorem concerns the number of real zeros that a polynomial function
may have. In counting the zeros of a polynomial, we count each zero as many
times as its multiplicity.
A polynomial function cannot have more real zeros than its degree.

The proof is based on the factor theorem. If r is a real zero of a polynomial
function f, then f(r)=0 and so, x-r is a factor of f(x). Each real zero
corresponds to a factor of degree 1. Because f cannot have more first degree
factors than its degree.

Descarte's rule of signs provides information about the number and location of
the real zeros of a polynomial function written in standard form. It requires
that we count the number of variations in the sign of the coefficients of f(x)
and f(-x). For example, the following polynomial function has two variations in
the signs of the coefficients.
\(f(x)=3x^7+4x^4+3x^2-2x-1 = -3x^7+0x^6+0x^5+4x^4+0x^3+3x^2-2x-1\)
Notice that we ignored the zero coefficients in \(0x^6,0x^5\) and \(0x^3\) in
counting the number of variations in the sign of f(x). Replacing x by -x, we
get:
\(f(-x)=-3(-x)^7+4(-x)^4+3(-x)^2-2(-x)-1 = 3x^7+4x^4+3x^2+2x-1\)
This has one variation in sign.

Descarte's Rule of Signs
Let f denote a polynomial function written in standard form. 
The number of positive real zeros of f either equals the number of variations in
the sign of the nonzero coefficients of f(x) or else equals that number  less an
even integer.
The number of negative real zeros of f either equals the number of variations in
the sign of the nonzero coefficients of f(-x) or else equals that number less an
even integer.

Example 3
Using the number of real zeros theorem and the rule of signs.
Discuss the real zeros of \(f(x)=3x^6-4x^4+3x^3+2x^2-x-3\)

Solution:
Because the polynomial is of degree 6, by the real zeros theorem there are at
most six real zeros. Since there are three variations in the sign of the nonzero
coefficients of f(x), by the rule of signs, we expect either three or one
positive real zero. To continue, look at f(-x).
\(f(-x)=3x^6-4x^4-3x^3+2x^2+x-3\)
There are three variations in sign, so we expect either three or one negative
real zeros. Equivalently, we know now that the graph of f has either three or one
positive x-intercepts and three or one negative x-intercepts.

Rational Zeros Theorem
The rational zeros theorem provides information about the rational zeros of a
polynomial with integer coefficients.
Let f be a polynomial function of degree 1 or higher of the form:
\(f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...a_{1}x+a_{0}\)
Each coefficient is an integer. If p/q is in lowest terms, it is a rational zero
of f, then p must be a factor of \(a_o\), and q must be a factor of \(a_n\).

Example 4
Listing potential rational zeros
\(f(x)=2x^3+11x^2-7x-6\)

Solution:
Because f has integer coefficients, we may use the rational zero theorem. First,
we list all the integers p that are factors of the constant term a0=-6 and all
the integers q that are factors of the leading coefficient ay=2.
\(p:....\pm1,\pm2,\pm3,\pm6\)
\(q:....\pm1,\pm2\)
Now, we form all possible ratios p/q
\(\frac{p}{q}:....\pm1,\pm2,\pm3,\pm6,\pm\frac{1}{2},\pm\frac{3}{2}\)
If f has a rational zero, it will be found in this list, which contains 12
possibilities.

Be sure that you understand what the rational zeros theorem says. For a
polynomial with integer coefficients, if there is a rational zero, it is one of
those listed. It may be the case that the function does not have any rational
zeros.

Long division, synthetic division, or substitution can be used to test each
potential rational zero to determine whether it is indeed a zero. To make the
work easier, integers are usually tested first.

Example 5
Finding the rational zeros of a polynomial function
\(f(x)=2x^3+11x^2-7x-6\)
Write f in factored form

Solution:
We gather all the information that we can about the zeros.
1. There are at most 3 zeros
2. By the rule of signs, there is one positive real zero. Also, because
\(f(-x)=-2x^3+11x^2+7x-6\)
There are two negative zeros or no negative zeros.
3. Now we use the list of potential rational zeros obtained in the previous
example.
\(\pm1,\pm2,\pm3,\pm6,\pm\frac{1}{2},\pm\frac{3}{2}\)
We choose to test the potential rational zero using substitution.
\(f(1)=2(1)^3+11(1)^2-7(1)-6 = 2+11-7-6 = 0\)
Since f(1)=0, 1 is a zero and x-1 is a factor of f. We can use long division or
synthetic division to factor f.
\(f(x)=2x^3+11x^2-7x-6 = (x-1)(2x^2+13x+6)\)

Now, any solution of the equation \(2x^2+13x+6=0\) will be a zero of f. Because
of this, we call the equation a depressed equation of f. Since the degree of the
depressed equation of f is less than that of the original polynomial, it is
easier to work with the depressed equation to find the zeros of f.
4. The depressed equation \(2x^2+13x+6=0\) is a quadratic equation with
discriminant \(b^2-4ac = 169-48 = 121 > 0\). The equation has two real
solutions, which can be found by factoring:
\(2x^2+13x+6 = (2x+1)(x+6) = 0\)
\(2x+1 = 0\) and \(x+6 = 0\)
So, \(x=-\frac{1}{2}\) or \(x = -6\)
The zeros of f are \(-6,-\frac{1}{2},1\)
We know that \(f(x)=2x^3+11x^2-7x-6\) can be written as
\(f(x)=(x-1)(2x^2+13x+6)\). Because \(2x^2+13x+6 = (2x+1)(x+6)\), we write f in
factored form as: \(f(x)=(x-1)(2x+1)(x+6)\)
Notice that the three zeros of f are found in this example are among those given
in the list of potential zeros. 

Find the Real Zeros of a Polynomial Function
1. Use the degree of the polynomial to determine the maximum number of real
zeros.
2. Use the rule of signs to determine the possible number of positive zeros and
negative zeros.
3. If the polynomial has integer coefficients, use the rational zeros theorem to
identify those rational numbers that potentially could be zeros.
Use substitution, synthetic division, or long division to test each potential
rational zero. Each time that a zero(and thus a factor) is found, repeat step 3
on the depressed equation.
4. In attempting to find the zeros, remember to use the factoring techniques
that you already know.

Example 6
Finding the real zeros of a polynomial function
Find the real zeros of \(f(x)=x^5-5x^4+12x^3-24x^2+32x-16\)
Write f in factored form.

Solution:
We gather all the information that we can about the zeros.
There are at most 5 real zeros because of \(x^5\).
By the rule of signs, there are 5, three, or one positive zero.
There are no negative zeros because \(f(x)=-x^5-5x^4-12x^3-24x^2-32x-16\) has no
variation in sign.
Because the leading coefficient \(a_{5}=1\) and there are no negative zeros, the
potential rational zeros are the integers 1,2,4,8, and 16, the positive factors
of the constant term, 16. We test the potential rational zero 1 first, using
synthetic division.

We get a remainder of f(1)=0, so 1 is a zero and x-1 is a factor of f. Using the
entries in the bottom row of the synthetic division, we can begin to factor f.
\(f(x)=x^5-5x^4+12x^3-24x^2+32x-16 = (x-1)(x^4-4x^3+8x^2-16x+16)\)
We now work with the first depressed equation:
\(q1(x)=x^4-4x^3+8x^2-16x+16=0\)
The potential rational zeros of q1 are still 1,2,4,8, and 16. We test 1 first
since it may be a repeated zero of f.
Since the remainder is 5, 1 is not a repeated zero. We try 2 next:
The remainder is f(2)=0 so 2 is a zero and x-2 is a factor of f. Again using the
bottom row, we find:
\(f(x)=x^5-5x^4+12x^3-24x^2+32x-16 = (x-1)(x-2)(x^3-2x^2+4x-8)\)
The remaining zeros satisfy the new depressed equation:
\(q2(x)=x^3-2x^2+4x-8=0\)
Notice that q2(x) can be factored using grouping.
Since \(x^2+4=0\) has no real solutions, the real zeros of f are 1 and 2, the
latter being a zero of multiplicity 2. The factored form of f is:
\(f(x)=x^5-5x^4+12x^3-24x^2+32x-16 = (x-1)(x-2)^2(x^2+4)\)

Example 7
Solving a polynomial equation
Find the real solutions of the equation:
\(x^5-5x^4+12x^3-24x^2+32x-16=0\)

Solution:
The real solutions of this equation are the real zeros of the polynomial
function \(f(x)=x^5-5x^4+12x^3-24x^2+32x-16\)
Using the result from previous example, the real zeros of f are 1 and 2. These
are the real solutions of the equation:
\(x^5-5x^4+12x^3-24x^2+32x-16=0\)

In example 6, the quadratic factor \(x^2+4\) that appears in the factored form
of f is called irreducible, because the polynomial \(x^2+4\) cannot be factored
over the real numbers. In general, we say that a quadratic  factor \(ax^2+bx+c\)
is irreducible if it cannot be factored over the real numbers, that is, if it is
prime over the real numbers.

Refer to examples 5 and 6. The polynomial function of example 5 has three real
zeros, and its factored form contains three linear factors. The polynomial
function of example 6 has two distinct real zeros, and its factored form
contains two distinct linear factors and one irreducible quadratic factor.

Every polynomial function with real coefficients can be uniquely factored into a
product of linear factors and/or irreducible quadratic factors.

We shall prove this result and we shall draw several additional conclusions
about the zeros of a polynomial function. One conclusion is worth noting. If a
polynomial with real coefficients is of odd degree, it must contain at least one
linear factor. This means it must have at least one real zero.

A polynomial function with real coefficients of odd degree has at least one real
zero.

Theorem for Bounds on Zeros
The search for the real zeros of a polynomial function can be reduced somewhat
if bounds on the zeros are found. A number M is a bound on the zeros of a
polynomial if every zero lies between -M and M. That is, M is a bound on the
zeros of a polynomial if \(-M \leq\text{ any real zero of f }\leq M\)

Example 8
Using the theorem for finding bounds on zeros
Find a bound on the real zeros of each polynomial
1. \(f(x)=x^5+3x^3-9x^2+5\)
2. \(g(x)=4x^5-2x^3+2x^2+1\)

Solution:
1. The leading coefficient of f is 1.
   \(f(x)=x^5+3x^3-9x^2+5 = 0,3,-9,0 = 5\)
   The smaller of the two numbers, 10, is the bound. Every real zero of f lies
   between -10 and 10.
2. First, we write g so that it is the product of a constant times a polynomial
   whose leading coefficient is 1.
   \(g(x)=4x^5-2x^3+2x^2+1 = 4(x^5-\frac{1}{2}x^3+\frac{1}{2}x^2+\frac{1}{4}\)
   Next, we evaluate the two expressions in (4) with
   \(a_4=0,a_3=-\frac{1}{2},a_2=\frac{1}{2},a_1=0\text{ and }a_0=\frac{1}{4}\)
   The smaller of the two numbers, \(\frac{5}{4}\) is the bound. Every real zero
   of g lies between \(-\frac{5}{4}\text{ and }\frac{5}{4}\)

Intermediate Value Theorem
It is based on the fact that the graph of a polynomial function is continuous,
that is, it contains no holes  or gaps. Let f denote a polynomial function. If
a<b and if f(a) and f(b) are of opposite sign, there is at least one real zero
of f between a and b.

Example 9
Using the intermediate value theorem to locate real zeros
Show that \(f(x)=x^5-x^3-1\) has a real zero between 1 and 2.

Solution:
We evaluate f at 1 and 2
\(f(1)=-1\)
\(f(2)=23\)
Because f(1)<0 and f(2)>0, it follows from the intermediate value theorem that f
has a zero between 1 and 2.

Let us look at the polynomial f of example 9 more closely. Based on the rule of
signs, f has exactly one positive real zero. Based on the rational zeros
theorem, 1 is the only potential positive rational zero. Since f(1) does not
equal 0, we conclude that the zero between 1 and 2 is irrational. We can use the
intermediate value theorem to approximate it.
1. Find two consecutive integers a and a+1 such that f has a zero between them.
2. Divide the interval into 10 equal subintervals.
3. Evaluate f at each endpoint of the subintervals until the intermediate value
theorem applies. This endpoint then contains a zero.
4. Repeat the process until the desired accuracy is achieved.

Example 10
Approximating the real zeros of a polynomial function
Find the positive zero of:
\(f(x)=x^5-x^3-1\) to two decimal places.

Solution:
From example 9, we know that the positive zero is between 1 and 2. We divide the
interval [1,2] into 10 equal subintervals.
[1,1.1][1.1,1.2][1.2,[1.3][1.3,1.4][1.4,1.5][1.5,1.6][1.6,1.7][1.7,1.8][1.9,1.9][1.9,2].
Now we find the value of f at each endpoint until the intermediate value theorem
applies.
\(f(x)=x^5-x^3-1\)
f(1.0)=-1  f(1.2)=-0.23968  f(1.1)=-0.72049  f(1.3)=0.51593
We can stop here and conclude that the zero is between 1.2 and 1.3. now we
divide the interval[1.2,1.3] into 10 equal subintervals and proceed to evaluate
f at each endpoint. We conclude that the zero lies between 1.23 and 1.24 and
correct to two decimal places, the zero is 1.23.

Complex Zeros

Complex Zeros
A variable in the complex number system is referred to as a complex variable. A
complex number r is called a complex zero of f if f(r)=0.

We have learned that some quadratic equations have no real solutions, but that
in the complex number system every quadratic equation has a solution, either
real or complex. 

Every complex polynomial function f(x) of degree \(n \geq 1\) has at least one
complex zero. 

Let \(f(x)=a_n^n+a_{n-1}x^{n-1}+...+a_1x+a_0\)
By the fundamental theorem of algebra, f has at least one zero, r1. Then, by the
factor theorem, x-r1 is a factor and \(f(x)=(x-r_1)q_1(x)\) where where q1(x) is
a complex polynomial of degree n-1 whose leading coefficient is an. Repeating
this argument n times, we arrive at \(f(x)=(x-r_1)(x-r_2)...(x-r_n)q_n(x)\)
where \(q_n(x)\) is a complex polynomial of degree n-n=0 whose leading
coefficient is \(a_n\). That is, \(q_n(x)=a_nx^n=a_n\), and so
\(f(x)=a_n(x-r_1)(x-r_2)...(x-r_n)\). We conclude that every complex polynomial
function f(x) of degree \(n \geq 1\) has exactly n zeros. 

Conjugate Pairs Theorem
We can use the fundamental theorem of algebra to obtain valuable information
about the complex zeros of polynomials whose coefficients are real numbers.

In other words, for polynomials whose coefficients are real numbers, the complex
zeros occur in conjugate pairs. This result should not be all that surprising
since the complex zeros of a quadratic function occurred in conjugate pairs.

The importance of this result should be clear. Once we know that, say, \(3+4i\)
is a zero of a polynomial with real coefficients, then we know that \(3-4i\) is
also a zero. This result has an important corollary. A polynomial f of odd
degree with real coefficients has at least one real zero.

For example, the polynomial \(f(x)=x^5-3x^4+4x^3-5\) has at least one zero that
is a real number, since f is of degree 5 (odd) and has real coefficients. 

Example 1
Using the conjugate pairs theorem
A polynomial f of degree 5 whose coefficients are real numbers has the zeros 1,
5i, and 1+i. Find the remaining two zeros.

Solution:
Since f has coefficients that are real numbers, complex zeros appear as
conjugate pairs. It follows that -5i, the conjugate of 5i, and 1-i, the
conjugate of 1+i, are the two remaining zeros.

Example 2
Finding a polynomial function whose zeros are given
Find a polynomial function f of degree 4 whose coefficients are real numbers
that has the zeros 1, 1, and -4+i.

Solution:
since -4+i is a zero, by the conjugate pairs theorem, -4-i must also be a zero
of f. Because of the factor theorem, if f(c)=0, then x-c is a factor of of f(x).
So we can now write f as:
\(f(x)=a(x-1)(x-1)[x-(-4+i)][x-(-4-i)]\) where a is any real number.
\(f(x)=a(x^2-2x+1)[x^2-(-4+i)x-(-4-i)x+(-4+i)(-4-i)]\)
\(a(x^2-2x+1)(x^2+4x-ix+4x+ix+16+4i-4i-i^2)\)
\(a(x^2-2x+1)(x^2+8x+17)\)
\(a(x^4+8x^3+17x^2-2x^3-16x^2-34x+x^2+8x+17)\)
\(a(x^4+6x^3+2x62-26x+17)\)


Every polynomial function with real coefficients can be uniquely factored over
the real numbers into a product of linear factors and/or irreducible quadratic
factors.

This second degree polynomial has real coefficients and is irreducible over the
real numbers. So, the factors of f are either linear or irreducible quadratic
factors. 

Example 3
Finding the complex zeros of a polynomial
\(f(x)=3x^4+5x^3+25x^2+45x-18\)

Solution:
1. The degree of f is 4, so f will have four complex zeros.
2. The rule of signs provides information about the real zeros. For this
polynomial, there is one positive real zero. There are three or one negative
real zeros because \(f(-x)=3x^4-5x^3+25x^2-45x-18\) has three variations in
sign. 
3. The rational zeros theorem provides information about the potential rational
zeros of polynomials with integer coefficients. For this polynomial, which has
integer coefficients, the potential rational zeros are:
\(\pm 1/3, \pm 2/3, \pm 1, \pm 2, \pm 6, \pm 9, \pm 18\)
Since f(-2)=0, then -2 is a zero and x+2 is a factor of f. the depressed
equation is:
\(3x^3-x^2+27x-9=0\)
Factor by grouping
\(x^2(3x-1)+9(3x-1)=0\)
Factor out the common factor 3x-1.
\((x^2+9)(3x-1)=0\)
\(x^2+9=0\)
\(3x-1=0\)
\(x^2=-9\)
\(x=1/3\)
\(x=-3i,3i,1/3\)
So the complex zeros are \(-3i.3i.-2,1/3\) 
Factored form of f is:
\(f(x)=3x^4+5x^3+25x^2=45x-18 = 3(x+3i)(x-3i)(x+2)(x-1/3)\)