Functions and Function Notation

Matthew Schurmann

Learning Objectives

In this section, you will:

Determine whether a relation represents a function.
Find the value of a function.
Determine whether a function is one-to-one.
Use the vertical line test to identify functions.
Identify the graphs of toolkit functions and key characteristics.

A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

Determining Whether a Relation Represents a Function

A relation is a set of ordered pairs. The set, which is denoted here as a list of ordered pairs inside curly brackets, $\{ ...... \}$ , consisting of the first components of each ordered pair is called the domain and the set consisting of the second 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.

$\{\left(1,2\right),\left(2,4\right),\left(3,6\right),\left(4,8\right),\left(5,10\right)\}$

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 value, or independent variable, and is often labeled with the lowercase letter $x$ . Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter $y$ .

A function $f$ is a relation that assigns a single element in the range to each element 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 values, 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 let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

$\{\left(\text{odd},1\right), \left(\text{even},2\right),\left(\text{odd},3\right),\left(\text{even},4\right),\left(\text{odd},5\right)\}$

Notice that each element in the domain, $\{ \text{even},\text{odd}\}$ 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 domain, $\{1,3,5\}$ 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.

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.

How To

Given a relationship between two quantities, determine whether the relationship is a function.

Identify the input values.
Identify the output values.
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.

Determining If Menu Price Lists Are Functions

The coffee shop menu, shown below consists of items and their prices.

Is price a function of the item?
Is the item a function of the price?

A menu of donut prices from a coffee shop where a plain donut is 💲1.49 and a jelly donut and chocolate donut are 💲1.99.

Show 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.

Therefore, the item is a not a function of price.

Try It

The table below lists the five greatest baseball players of all time in order of rank.

Player	Rank
Babe Ruth	1
Willie Mays	2
Ty Cobb	3
Walter Johnson	4
Hank Aaron	5

Is the rank a function of the player name?
Is the player name a function of the rank?

Show Solution

a. yes;
b. yes. (Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)

Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into graphing calculators and 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 just as we use $x,y,$ and $z$ to represent numbers and $A,B,$
and $C$ to represent sets.

$\begin{align*} \text{height is } f \text{ of } a &\quad \text{We name the function } f; \text{ height is a function of age.} \\ h = f(a) &\quad \text{We use parentheses to indicate the function input.} \\ f(a) &\quad \text{We name the function } f; \text{ the expression is read as ``} f \text{ of } a\text{.''} \end{align*}$

Remember, we can use any letter to name the function; the notation $h\left(a\right)$ 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 $f\left(a+b\right)$ means “first add 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.

Function Notation

The notation $y=f \left(x\right)$ 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 $f \left(x\right)$ represents the output value, or dependent variable.

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.

Show 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 $\text{days}=f \left(\text{month}\right)$
or $d=f\left(m\right)$ . The name of the month is the input to a “rule” that associates a specific number (the output) with each input.

The function 31 = f(January) where 31 is the output, f is the rule, and January is the input.

For example, $f\left(\text{March}\right)=31$ , because March has 31 days. The notation $d=f\left(m\right)$ reminds us that the number of days, $d$ (the output), is dependent on the name of the month $m$ (the input).

Analysis

Note that the inputs to a function do not have to be numbers; function inputs can be 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.

Interpreting Function Notation

A function $N=f\left(y\right)$ gives the number of police officers $N$ in a town in year $y$ . What does $f\left(2005\right)=300$ represent?

Show Solution

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

Try It

Use function notation to express the weight of a pig in pounds as a function of its age in days $d$ .

Show Solution

$w=f\left(d\right)$

Instead of a notation such as $y=f\left(x\right)$ could we use the same symbol for the output as for the function, such as $,y=y\left(x\right)$ meaning “ $y$ is a function of $x$ ?”

Yes, this is often done, especially in applied subjects that use higher math, such as physics and engineering. However, in exploring math itself we like to maintain a distinction between a function such as $f$ which is a rule or procedure, and the output $y$ we get by applying $f$ to a particular input $x$ . This is why we usually use notation such as $y=f\left(x\right)$ , $P = W\left(d\right)$ , and so on.

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.

The table below has the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function $f$ where $D=f\left(m\right)$ identifies months by an integer rather than by name.

Month number $m$ (input)	1	2	3	4	5	6	7	8	9	10	11	12
Days in month $D$ (output)	31	28	31	30	31	30	31	31	30	31	30	31

The next table displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.

Age in years $y$ (input)	5	5	6	7	8	9	10
Height in inches $H$ (output)	40	42	44	47	50	52	54

How To

Given a table of input and output values, determine whether the table represents a function.

Identify the input and output values.
Check to see if each input value is paired with only one output value. If so, the table represents a function.

Identifying Tables that Represent Functions

Which table, A, B, or C, represents a function (if any)?

Table A

Input	Output
2	1
5	3
8	6

Table B

Input	Output
–3	5
0	1
4	5

Table C

Input	Output
1	0
5	2
5	4

Show Solution

Table A and Table B define functions. In both, each input value corresponds to exactly one output value. Table C does not define a function because the input value of 5 corresponds to two different output values.

When a table represents a function, corresponding input and output values can also be specified using function notation.

The function represented by Table A can be represented by writing

$f\left(2\right)=1, f\left(5\right)=3, \text{and }f\left(8right)=6$

Similarly, the statements

$g\left(-3\right)=5, g\left(0\right)=1, \text{and }g\left(4right)=5$

represent the function in Table B.

Table C cannot be expressed in a similar way because it does not represent a function.

Try It

Does the table below represent a function?

Input	Output
1	10
2	100
3	1000

Show Solution

yes

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 value 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.

Evaluation of Functions in Algebraic Forms

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

How To

Given the formula for a function, evaluate.

Replace the input variable in the formula with the value provided.
Calculate the result.

Evaluating Functions at Specific Values

Evaluate $f\left(x\right)={x}^{2}+3x-4$ at

$2$
$a$
$a+h$
$\dfrac{f\left(a+h\right)-f\left(a\right)}{h}$

Show Solution

Replace the $x$ in the function with each specified value.

Because the input value is a number, 2, we can use simple algebra to simplify.
$\vspace{5pt}$

$\begin{align*} f(2) &= 2^2 + 3(2) - 4 \\ &= 4 + 6 - 4 \\ &= 6 \end{align*}$
In this case, the input value is a letter so we cannot simplify the answer any further.
$\vspace{5pt}$

$f\left(a\right)={a}^{2}+3a-4$
$\vspace{5pt}$
With an input value of $a+h$ we must use the distributive property.
$\vspace{5pt}$

$\begin{align*} f(a + h) &= (a + h)^2 + 3(a + h) - 4 \\ &= a^2 + 2ah + h^2 + 3a + 3h - 4 \end{align*}$

$\vspace{5pt}$
In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that
$\vspace{5pt}$

$f\left(a+h\right)={a}^{2}+2ah+{h}^{2}+3a+3h-4$
$\vspace{5pt}$

and we know that
$\vspace{5pt}$

$f\left(a\right)={a}^{2}+3a-4$
$\vspace{5pt}$

Now we combine the results and simplify.
$\vspace{5pt}$

$\begin{align*} \frac{f(a + h) - f(a)}{h} &= \frac{\left(a^2 + 2ah + h^2 + 3a + 3h - 4\right) - \left(a^2 + 3a - 4\right)}{h} \\ &= \frac{2ah + h^2 + 3h}{h} \\ &= \frac{h(2a + h + 3)}{h} \quad \text{Factor out } h \\ &= 2a + h + 3 \quad \text{Simplify} \end{align*}$

Evaluating Functions

Given the function $h\left(p\right)={p}^{2}+2p$ evaluate $h\left(4\right)$ .

Show Solution

To evaluate $h\left(4\right)$ we substitute the value 4 for the input variable $p$ in the given function.

$\begin{align*} h(p) &= p^2 + 2p \\ h(4) &= (4)^2 + 2(4) \\ &= 16 + 8 \\ &= 24 \end{align*}$

Therefore, for an input of 4, we have an output of 24.

Finding Function Values from a Graph

Evaluating a function using a graph also requires finding the corresponding output 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 value(s).

Reading Function Values from a Graph

Given the graph in below,

Evaluate $f\left(2\right)$ .
Solve $f\left(x\right)=4$ .

Graph of a positive parabola centered at (1, 0).

Show Solution

To evaluate $f\left(2\right)$ locate the point on the curve where $x=2$ then read the y-coordinate of that point. The point has coordinates $\left(2,1\right)$ so $f\left(2\right)=1$ .
To solve $f\left(x\right)=4$ 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 $4:$ $\left(-1,4\right)$ and $\left(3,4\right)$ . These points represent the two solutions to $f\left(x\right)=4:$ $-1,$ or $,3.,$ This means $f\left(-1\right)=4$ , and $f\left(3\right)=4$ , 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 at the beginning of this chapter, the stock price was $\mathdollar 1000$ on five different dates, meaning that there were five different input values that all resulted in the same output value of $\mathdollar 1000$ .

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, as listed in the table below.

Letter grade	Grade point average
A	4.0
B	3.0
C	2.0
D	1.0

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.

One-to-One Function

A one-to-one function is a function in which each output value corresponds to exactly one input value. There are no repeated x– or y-values.

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?

Show Solution

A circle of radius $r$ has a unique area measure given by $A=\pi {r}^{2}$ so for any input, $r$ there is only one output, $A$ . 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: $\sqrt{\dfrac{A}{\pi}}$ . So the area of a circle is a one-to-one function of the circle’s radius.

Try It

Is a balance a function of the bank account number?
Is a bank account number a function of the balance?
Is a balance a one-to-one function of the bank account number?

Show Solution

a. yes, because each bank account has a single balance at any given time; b. no, because several bank account numbers may have the same balance; c. no, because the same output may correspond to more than one input.

Try It

If each percent grade earned in a course translates to one letter grade, is the letter grade a function of the percent grade?
If so, is the function one-to-one?

Show Solution

Yes, letter grade is a function of percent grade;
No, it is not one-to-one. There are 100 different percent numbers we could get but only about five possible letter grades, so there cannot be only one percent number that corresponds to each letter grade.

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 $x$ and the output value $y$ and we say $y$ is a function of $x$ or $y=f\left(x\right)$ when the function is named $f$ . The graph of the function is the set of all points $\left(x,y\right)$ in the plane that satisfies the equation $y=f\left(x\right)$ . If the function is defined for only a few input values, then the graph of the function consists of 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 below tell us that $f\left(0\right)=2$ and $f\left(6\right)=1$ . However, the set of all points $\left(x,y\right)$ satisfying $y=f\left(x\right)$ is a curve. The curve shown includes $\left(0,2\right)$ and $\left(6,1\right)$ because the curve passes through those points.

Graph of a polynomial.

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.

Three graphs visually showing what is and is not a function.

How To

Given a graph, use the vertical line test to determine if the graph represents a function.

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

Applying the Vertical Line Test

Which of the graphs in below represent(s) a function $y=f\left(x\right)$ ?

Graph of a polynomial.

Show 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, as shown below.

Graph of a circle.

Try It

Does the graph in below represent a function?

Graph of absolute value function.

Show Solution

yes

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.

How To

Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.

Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.
If there is any such line, determine that the function is not one-to-one.

Applying the Horizontal Line Test

Consider the graphs shown below. Determine which represent one-to-one functions.

Graph of a polynomial.

Show Solution

The graph (a) represents a function since it satisfies the vertical line test. The function is not one-to-one. The horizontal line shown before intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)

The graph in (b) represents a function since it satisfies the vertical line test. The function is one-to-one. Any horizontal line will intersect a diagonal line at most once.

The graph in (c) does not represent a functions since it does not satisfy the vertical line test. As a result, this graph does not represent a one-to-one function.

Identifying Basic Toolkit Functions

Throughout this course, 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 know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use $x$ as the input variable and $y=f\left(x\right)$ 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 characteristics. The graphs, properties, and key points are included with each function below.

Identity (Linear Function) $f\left(x\right)=x$ Domain: $(-\infty,\infty)$ Range: $(-\infty,\infty)$ x-intercept: $(0,0)$ y-intercept: $(0,0)$ Vertical Asymptote: None Horizontal Asymptote: None Key Points: $(-2,2), (-1,-1), (0,0), (1,1), (2,2)$
Base 10 (Exponential Function) $f\left(x\right)={10}^{x}$ Domain: $(-\infty,\infty)$ Range: $(0,\infty)$ x-intercept: None y-intercept: $(0,1)$ Vertical Asymptote: None Horizontal Asymptote: $y=0$ Key Points: $(0,1), (1,10)$
Base $e$ (Exponential Function) $f\left(x\right)={e}^{x}$ Domain: $(-\infty,\infty)$ Range: $(0,\infty)$ x-intercept: None y-intercept: $(0,1)$ Vertical Asymptote: None Horizontal Asymptote: $y=0$ Key Points: $(0,1), (1,e)$
Common Logarithm (Logarithmic Function) $f\left(x\right)=\text{log}(x)$ Domain: $(0,\infty)$ Range: $(-\infty,\infty)$ x-intercept: $(1,0)$ y-intercept: None Vertical Asymptote: $x=0$ Horizontal Asymptote: None Key Points: $(1,0), (10,1)$
Quadratic (Polynomial Function) $f\left(x\right)={x}^{2}$ Domain: $(-\infty,\infty)$ Range: $[0,\infty)$ x-intercept: $(0,0)$ y-intercept: $(0,0)$ Vertical Asymptote: None Horizontal Asymptote: None Key Points: $(-2,4), (-1,1), (0,0), (1,1), (2,4)$
Cubic (Polynomial Function) $f\left(x\right)={x}^{3}$ Domain: $(-\infty,\infty)$ Range: $(-\infty,\infty)$ x-intercept: $(0,0)$ y-intercept: $(0,0)$ Vertical Asymptote: None Horizontal Asymptote: None Key Points: $(-2,-8), (-1,-1), (0,0), (1,1), (2,8)$
Reciprocal (Rational Function) $f\left(x\right)=\dfrac{1}{x}$ Domain: $(-\infty,0) \bigcup (0,\infty)$ Range: $(-\infty,0) \bigcup (0,\infty)$ x-intercept: None y-intercept: None Vertical Asymptote: $x=0$ Horizontal Asymptote: $y=0$ Key Points: $\left(-2,-\dfrac{1}{2}\right), (-1,-1), (1,1), \left(2,\dfrac{1}{2}\right)$
Reciprocal squared (Rational Function) $f\left(x\right)=\dfrac{1}{{x}^{2}}$ Domain: $(-\infty,0) \bigcup (0,\infty)$ Range: $(0,\infty)$ x-intercept: None y-intercept: None Vertical Asymptote: $x=0$ Horizontal Asymptote: $y=0$ Key Points: $\left(-2,-\dfrac{1}{4}\right), (-1,-1), (1,1), \left(2,\dfrac{1}{4}\right)$
Square root (Power Function) $f\left(x\right)=\sqrt{x}$ $f\left(x\right)={x}^{\frac{1}{2}}$ Domain: $[0,\infty)$ Range: $[0,\infty)$ x-intercept: $(0,0)$ y-intercept: $(0,0)$ Vertical Asymptote: None Horizontal Asymptote: None Key Points: $(1,1), (4,2), (9,3)$
Cube root (Power Function) $f\left(x\right)=\sqrt[3]{x}$ $f\left(x\right)={x}^{\frac{1}{3}}$ Domain: $(-\infty,\infty)$ Range: $(-\infty,\infty)$ x-intercept: $(0,0)$ y-intercept: $(0,0)$ Vertical Asymptote: None Horizontal Asymptote: None Key Points: $(-8,-2), (-1,-1), (0,0), (1,1), (8,2)$

Access the following online resources for additional instruction and practice with functions.

Key Concepts

A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output.
Function notation is a shorthand method for relating the input to the output in the form $y=f\left(x\right)$ .
In tabular form, a function can be represented by rows or columns that relate to input and output values.
To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value.
To solve for a specific function value, we determine the input values that yield the specific output value.
An algebraic form of a function can be written from an equation.
Input and output values of a function can be identified from a table.
Relating input values to output values on a graph is another way to evaluate a function.
A function is one-to-one if each output value corresponds to only one input value.
A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.
The graph of a one-to-one function passes the horizontal line test.

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Learning Objectives

Determining Whether a Relation Represents a Function

Function

How To

Determining If Menu Price Lists Are Functions

Try It

Using Function Notation

Function Notation

Using Function Notation for Days in a Month

Analysis

Interpreting Function Notation

Try It

Representing Functions Using Tables

How To

Identifying Tables that Represent Functions

Try It

Finding Input and Output Values of a Function

Evaluation of Functions in Algebraic Forms

How To

Evaluating Functions at Specific Values

Evaluating Functions

Finding Function Values from a Graph

Reading Function Values from a Graph

Determining Whether a Function is One-to-One

One-to-One Function

Determining Whether a Relationship Is a One-to-One Function

Try It

Try It

Using the Vertical Line Test

How To

Applying the Vertical Line Test

Try It

Using the Horizontal Line Test

How To

Applying the Horizontal Line Test

Identifying Basic Toolkit Functions

Key Concepts

License

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