# 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.