Functions and Notation in Precalculus
These are my notes on functions and notation in precalculus.
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.