Donna Kirk

A rear view of a student raising her hand as she sits at her desk. A teacher standing at the front of the classroom is pointing at the student.

Figure 5.55 A small group of elementary students learning from their teacher. (credit: modification of work "Our school" by Woodleywonderworks/Flickr, CC BY 2.0 )

Learning Objectives

After completing this section, you should be able to:

Use function notation.
Determine if a relation is a function with different representations.
Apply the vertical line test.
Determine the domain and range of a function.

In this section, we will learn about relations and functions. As we go about our daily lives, we have many data items or quantities that are paired to our names. Our social security number, student ID number, email address, phone number, and our birthday are matched to our name. There is a relationship between our name and each of those items. When your teacher gets their class roster, the names of all the students in the class are listed in one column and then the student ID number is likely to be in the next column. If we think of the correspondence as a set of ordered pairs, where the first element is a student name and the second element is that student’s ID number, we call this a relation.

(Student name, Student ID #)

The set of all the names of the students in the class is called the domain of the relation and the set of all student ID numbers paired with these students is the range of the relation. In general terms, a relation is any set of ordered pairs, ( $x, y$ ). All the $x$ -values in the ordered pairs together make up the domain. All the $y$ -values in the ordered pairs together make up the range.

There are many situations similar to the student's name and student ID # where one variable is paired or matched with another. The set of ordered pairs that records this matching is a relation. A special type of relation, called a function, occurs extensively in mathematics. A function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each $x$ -value is matched with only one $y$ -value.

Let us look at the relation between your friends and their birthdays in Figure 5.56. Every friend has a birthday, but no one has two birthdays. It is okay for two people to share a birthday. It is okay that Danny and Stephen share July 24 as their birthday and that June and Liz share August 2. Since each person has exactly one birthday, the relation is a function.

Mapping of names and birthdays. Names and birthdays are as follows. Alison: April 25. Penelope: May 23. June: August 2. Gregory: September 15. Geoffrey: January 12. Lauren: May 10. Stephen: July 24. Alice: February 3. Liz: August 2. Danny: July 24.

Figure 5.56 Birthday Mapping

Use Function Notation

It is very convenient to name a function; most often functions are named $f$ , $g$ , $h$ , $F$ , $G$ , or $H$ . In any function, for each $x$ -value from the domain, we get a corresponding $y$ -value in the range. In the function $f$ , we write this range value $y$ as $f$ ( $x$ ). This notation $f$ ( $x$ ) is called function notation and is read "f of $x$ " or "the value of f at $x$ ." In this case the parentheses do not indicate multiplication.

We call $x$ the independent variable as it can be any value in the domain. We call $y$ the dependent variable as its value depends on $x$ . Much like when you first encountered the variable $x$ , function notation may be rather unsettling. But the more you use the notation, the more familiar you become with the notation, and the more comfortable you will be with it.

Let’s review the equation $y = 4 x - 5$ . To find the value of $y$ when $x = 2$ , we know to substitute $x = 2$ into the equation and then simplify.

	$y = 4 x - 5$
Let $x = 2$ .	$\begin{matrix} y & = & 4 \cdot 2 - 5 \\ y & = & 3 \end{matrix}$

The value of the function at $x = 2$ is 3. We do the same thing using function notation, the equation $y = 4 x - 5$ can be written as $f (x) = 4 x - 5$ . To find the value when $x = 2$ , we write:

	$f (x) = 4 x - 5$
Let $x = 2$ .	$\begin{matrix} f (2) & = & 4 \cdot 2 - 5 \\ f (2) & = & 3 \end{matrix}$

The value of the function at $x = 2$ is 3. This process of finding the value of $f (x)$ for a given value of $x$ is called evaluating the function.

Example 5.61

Evaluating the Function

For the function $f (x) = 2 x^{2} + 3 x - 1$ , evaluate the function.

$f (3)$
$f (- 2)$
$f (a)$

Solution

To evaluate $f (3)$ , substitute 3, for $x$ .
Simplify.
$\begin{matrix} f (x) & = & 2 x^{2} + 3 x - 1 \\ f (3) & = & 2 {(3)}^{2} + 3 \cdot 3 - 1 \\ f (3) & = & 2 \cdot 9 + 3 \cdot 3 - 1 \\ f (3) & = & 18 + 9 - 1 \\ f (3) & = & 26 \end{matrix}$
To evaluate $f (- 2)$ , substitute $- 2$ for $x$ .
Simplify.
$\begin{matrix} f (x) & = & 2 x^{2} + 3 x - 1 \\ f (- 2) & = & 2 {(- 2)}^{2} + 3 (- 2) - 1 \\ f (- 2) & = & 2 \cdot 4 + (- 6) - 1 \\ f (- 2) & = & 8 + (- 6) - 1 \\ f (- 2) & = & 1 \end{matrix}$
To evaluate $f (a)$ , substitute $a$ for $x$ .
Simplify.
$\begin{matrix} f (x) & = & 2 x^{2} + 3 x - 1 \\ f (a) & = & 2 {(a)}^{2} + 3 \cdot a - 1 \\ f (a) & = & 2 a^{2} + 3 a - 1 \end{matrix}$

Your Turn 5.61

For the function

f\left( x \right) = 3{x^2} - 2x + 1

, evaluate the function.

1.

f\left( 3 \right)

2.

f\left( { - 1} \right)

3.

f\left( t \right)

Example 5.62

Evaluating the Function in an Application

The number of unread emails in Sylvia’s inbox is 75. This number grows by 10 unread emails a day. The function $N (t) = 75 + 10 t$ represents the relation between the number of emails, $N$ , and the time, $t$ , measured in days. Find $N$ (5). Explain what this result means.

Solution

Find $N$ (5). Explain what this result means.

Substitute in $t = 5$ .
Simplify.

\begin{matrix} N (5) & = & 75 + 10 \cdot 5 \\ N (t) & = & 75 + 10 t \\ N (5) & = & 75 + 50 \\ N (5) & = & 125 \end{matrix}

If 5 is the number of days, $N (5)$ is the number of unread emails after 5 days. After 5 days, there are 125 unread emails in Sylvia’s inbox.

Your Turn 5.62

1.

The number of unread emails in Bryan’s account is 100. This number grows by 15 unread emails a day. The function

N\left( t \right) = 100 + 15t

represents the relation between the number of emails,

N

, and the time,

t

, measured in days. Find

N

(7). Explain what the result means.

Determining If a Relation Is a Function with Different Representations

We can determine whether a relation is a function by identifying the input and the output values. If each input value leads to only one output value, classify the relation as a function. If any input value leads to two or more outputs, do not classify the relation as a function.

We will review three different representations of relations and determine if they are functions: ordered pairs, mapping, and equations.

Example 5.63

Determining If a Relation Is a Function with a Set of Ordered Pairs

Use the set of ordered pairs to determine whether the relation is a function.

${(- 3, 27), (- 2, 8), (- 1, 1), (0, 0), (1, 1), (2, 8), (3, 27)}$
${(9, - 3), (4, - 2), (1, - 1), (0, 0), (1, 1), (4, 2), (9, 3)}$

Solution

Each $x$ -value is matched with only one $y$ -value. This relation is a function.
The $x$ -value 9 is matched with two $y$ -values, both 3 and $- 3$ . This relation is not a function.

Your Turn 5.63

Use the set of ordered pairs to determine whether the relation is a function.

1.

\left\{ {\left( {-3,-6} \right),\left( {-2,-4} \right),\left( {-1,-2} \right),\left( {0,0} \right),\left( {1,2} \right),\left( {2,4} \right),\left( {3,6} \right)} \right\}

2.

{\text{\{ (8, - 4), (4, - 2), (2, - 1), (0,0), (2,1), (4,2), (8,4)\} }}

A mapping is sometimes used to show a relation. The arrows show the pairing of the elements of the domain with the elements of the range. Consider the example of the relation between your friends and their birthdays used in Figure 5.57. In this particular example, the domain is the set of people’s names, and the range is the set of their birthdays. This mapping was a function because everybody’s name maps to exactly one birthday.

Example 5.64

Determining If a Relation Is a Function with Mapping

Use the mapping in Figure 5.57 to determine whether the relation is a function.

Mapping of names and phone numbers. The names and corresponding phone numbers are as follows. Lydia: 321-549-3327 home and 321-964-7324 cell. Eugene: 427-658-2314 cell. Janet: 427-658-2314 cell. Rick: 798-367-8541 cell. Marty: 684-358-7961 home and 684-369-7231 cell.

Figure 5.57

Solution

Both Lydia and Marty have two phone numbers. Each $x$ -value is not matched with only one $y$ -value. This relation is not a function.

Your Turn 5.64

1.

Use the mapping in the given figure to determine whether the relation is a function.

Mapping of names and phone numbers. The names and corresponding phone numbers are as follows. Neal: 753-469-9731 cell. Krystal: 684-369-7231 cell. Kelvin: 123-567-4839 work. George: 123-567-4839 work and 639-847-6971 cell. Christa: 567-534-2970 work. Mike: 798-367-8541 cell and 567-534-2970 work.

In algebra functions will usually be represented by an equation. It is easiest to see if the equation is a function when it is solved for $y$ . If each value of $x$ results in only one value of $y$ , then the equation defines a function.

Example 5.65

Determining If a Relation Is a Function with an Equation

Determine whether each equation is a function. Assume $x$ is the independent variable.

$2 x + y = 7$
$y = x^{2} + 1$
$x + y^{2} = 3$

Solution

$2 x + y = 7$
For each value of $x$ , we multiply it by $- 2$ and then add 7 to get the $y$ -value.

For example, if $x = 3$ :

$\begin{matrix} y & = & - 2 x + 7 \\ y & = & - 2 \cdot 3 + 7 \\ y & = & 1 \end{matrix}$

We have that when $x = 3$ , then $y = 1$ . It would work similarly for any value of $x$ . Since each value of $x$ , corresponds to only one value of $y$ the equation defines a function.
$y = x^{2} + 1$
For each value of $x$ , we square it and then add 1 to get the $y$ -value.

For example, if $x = 2$

$\begin{matrix} y & = & x^{2} + 1 \\ y & = & 2^{2} + 1 \\ y & = & 5 \end{matrix}$

We have that when $x = 2$ , then $y = 5$ . It would work similarly for any value of $x$ . Since each value of $x$ corresponds to only one value of $y$ , the equation defines a function.
$x + y^{2} = 3$
$x + y^{2} = 3$

Isolate the $y$ term.

$y^{2} = - x + 3$

Let us substitute $x = 2$ .

$y^{2} = - 2 + 3$

$y^{2} = 1$

This gives us two values for $y$ .

$y = 1, y = - 1$

We have shown that when $x = 2$ , then $y = 1$ and $y = - 1$ . It would work similarly for any value of $x$ . Since each value of $x$ does not corresponds to only one value of $y$ the equation does not define a function.

Your Turn 5.65

Determine whether each equation is a function.

1.

4x + y = - 3

2.

x + {y^2} = 1

3.

y-{x^2} = 2

Video

Relations and Functions

Applying the Vertical Line Test

We reviewed how to determine if a relation is a function. The relations we looked at were expressed as a set of ordered pairs, a mapping, or an equation. We will now cover how to tell if a graph is that of a function.

An ordered pair $(x, y)$ is a solution of a linear equation, if the equation is a true statement when the $x$ -values and $y$ -values of the ordered pair are substituted into the equation. The graph of a linear equation is a straight line where every point on the line is a solution of the equation, and every solution of this equation is a point on this line. Figure 5.58 we can see that in the graph of the equation $y = 2 x - 3$ , for every $x$ -value there is only one $y$ -value, as shown in the accompanying table.

An x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. A line representing y equals 2 x minus 3 passes through the points, (negative 3, negative 9), (0, negative 3), (1.5, 0), and (6, 9). Five vertical dashed lines are present. The first line lies between the points, (negative 2, 0) and (negative 2, negative 7). The second line lies between the points, (negative 1, 0) and (negative 1, negative 5). The third line lies between the points, (0, 0) and (0, negative 3). The fourth line lies between the points, (3, 0) and (3, 3). The fifth line lies between the points, (4, 0) and (4, 5).

Figure 5.58 Graph of the Equation

y = 2 x - 3

A relation is a function if every element of the domain has exactly one value in the range. The relation defined by the equation $y = 2 x - 3$ is a function. If we look at the graph, each vertical dashed line only intersects the solid line at one point. This makes sense as in a function, for every $x$ -value there is only one $y$ -value. If the vertical line hit the graph twice, the $x$ -value would be mapped to two $y$ -values, and so the graph would not represent a function. This leads us a graphical method of determining functions called the vertical line test, which states that a set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. If any vertical line intersects the graph in more than one point, the graph does not represent a function.

Example 5.66

Applying the Vertical Line Test

Determine whether the graph (Figure 5.59) is the graph of a function applying the vertical line test.

A line is plotted on an x y coordinate plane. The x and y axes range from negative 6 to 6, in increments of 1. The line passes through the points, (negative 5, 5), (0, 2), and (5, negative 1). Note: all values are approximate.

Figure 5.59

Solution

On the graph (Figure 5.60), only three vertical dashed lines are drawn. However, it can be determined that any vertical dashed line that is drawn will intersect the solid line at exactly one point. It is the graph of a function.

Figure 5.60

Your Turn 5.66

1.

Determine whether the graph is the graph of a function.

A parabola is plotted on an x y coordinate plane. The x-axis ranges from negative 6 to 6, in increments of 1. The y-axis ranges from negative 2 to 10, in increments of 1. The parabola opens up and it passes through the points, (negative 3, 8), (negative 2, 3), (negative 1, 0), (0, negative 1), (1, 0), (2, 3), and (3, 8).Note: all values are approximate.

Example 5.67

Applying the Vertical Line Test to a Parabola

Determine whether the graph is the graph of a function (Figure 5.61).

A parabola is plotted on an x y coordinate plane. The x and y axes range from negative 6 to 6, in increments of 1. The parabola opens to the right and it passes through the points, (3, 2), (0, 1), (negative 1, 0), (0, negative 1), and (3, negative 2). Note: all values are approximate.

Figure 5.61

Solution

Figure 5.62 does not represent a function since the vertical dashed lines shown on the graph below intersect the solid line at two points.

Figure 5.62

Your Turn 5.67

1.

Determine whether the graph is the graph of a function.

An ellipse is plotted on an x y coordinate plane. The x and y axes range from negative 6 to 6, in increments of 1. The center of the ellipse is at (0, 0). The ellipse passes through the points, (negative 2, 0), (0, 3), (2, 0), and (0, negative 3).

Determining the Domain and Range of a Function

For the function $y = f (x), x$ is the independent variable as it can be any value in the domain, and $y$ is the dependent variable since its value depends on $x$ . For the function $y = f (x)$ , the values of $x$ make up the domain and the values of $y$ make up the range.

Example 5.68

Finding the Domain and Range of Ordered Pairs

For ${(1, 1), (2, 4), (3, 9), (4, 16), (5, 25)}$ :

Find the domain of the relation.
Find the range of the relation.

Solution

The domain is the set of all $x$ -values of the relation: ${1, 2, 3, 4, 5}$
The range is the set of all $y$ -values of the relation: ${1, 4, 9, 16, 25}$

Your Turn 5.68

For the relation

\left\{ {\left( {1,1} \right),\left( {2,8} \right),\left( {3,27} \right),\left( {4,64} \right),\left( {5,125} \right)} \right\}

:

1.

Find the domain of the relation.

2.

Find the range of the relation.

Example 5.69

Finding the Domain and Range on a Graph

Use Figure 5.63 to:

List the ordered pairs of the relation.
Find the domain of the relation.
Find the range of the relation.

Six points are plotted on an x y coordinate plane. The x and y axes range from negative 6 to 6, in increments of 1. The points are plotted at the following coordinates: (negative 3, negative 1), (negative 3, 4), (0, 3), (1, 5), (2, negative 2), and (4, negative 2).

Figure 5.63

Solution

The ordered pairs of the relation are: ${(1, 5), (- 3, - 1), (4, - 2), (0, 3), (2, - 2), (- 3, 4)}$ .
The domain is the set of all $x$ -values of the relation: ${- 3, 0, 1, 2, 4}$ . Notice that while $- 3$ repeats, it is only listed once.
The range is the set of all $y$ -values of the relation: ${- 2, - 1, 3, 4, 5} .$ Notice that while $- 2$ repeats, it is only listed once.

Your Turn 5.69

Use the given figure to:

1.

List the ordered pairs of the relation.

2.

Find the domain of the relation.

3.

Find the range of the relation.

Video

Domain and Range on Graphs

Who Knew?

Function and Function Notation

In 1673, Gottfried Leibniz, the German mathematician who co-invented calculus, seems to be the first person to use the word function in a mathematical sense, although his use of it does not exactly fit with the modern use and definition. The person who is credited with the modern definition of function is Swiss mathematician Johann Bernoulli, who wrote about it in a letter to Leibniz in 1698. Supposedly, Leibniz wrote Bernoulli back, approving of this use of the word. In 1734, the use of the notation $f (x)$ for a function was first used by Swiss mathematician Leonhard Euler (pronounced “Oiler”). Euler had a knack for inventing notation. He also introduced the notation $e$ for the base of natural logs (1727), $i$ for the square root of $- 1$ (1777), $\sum$ for summation (1755), and many others. Euler also introduced many other ideas associated with functions. Euler defined exponential functions and defined logarithmic functions as their inverse; he also introduced the beta and gamma functions, and was the first person to consider the trigonometric identities (sine, cosine, etc.) as functions.

Check Your Understanding

57.

If

f\left( x \right) = 2x-8

then

f\left(3\right) = -2

.

True
False

58.

\left\{ {\left( {1,2} \right),\left( {2,3} \right),\left( {3,4} \right),\left( {2,1} \right),\left( {3,2} \right),\left( {4,3} \right)} \right\}

represent the ordered pairs of a function.

True
False

59.

The graph shown represents the graph of a function:

Two functions are graphed on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. The first function passes through the points, (negative 6, 9), (negative 2, 4.5), (1, 0), (negative 2, negative 4.5), and (negative 6, negative 9). The second function passes through the points, (9, 6), (7, 3.5), (5, 0), (7, negative 3.5), and (9, negative 6). Note: all values are approximate.

True
False

60.

The figure shown represents the mapping of a function.
Mapping of two sets of values. Mapping infers the following data: 10, 15; 20, 45; 30, 25; and 40, 35.

True
False

61.

The domain of the mapping in the figure is

\left\{ {15,25,35,45} \right\}

.

True
False

Section 5.7 Exercises

For the following exercises, evaluate the functions at the values

f\left({-2}\right),f\left({-1}\right),f\left(0\right),f\left(1\right),

and

f\left(2\right)

.

1 .

f\left(x\right)=4-2x

2 .

f\left(x\right)=8-3x

3 .

f\left( x \right) = 8{x^2}-7x + 3

4 .

f\left( x \right) = 3 + \sqrt {x + 3}

5 .

f\left( x \right) = \frac{{x - 2}}{{x + 3}}

6 .

f\left( x \right) = {x^3}

For the following exercises, determine whether the ordered pairs represent a function.

7 .

\left\{ {\left( {-1,-1} \right),\left( {-2,-2} \right),\left( {-3,-3} \right)} \right\}

8 .

\left\{ {\left( {3,4} \right),\left( {4,5} \right),\left( {5,6} \right)} \right\}

9 .

\left\{ {\left( {2,5} \right),\left( {7,11} \right),\left( {15,8} \right),\left( {7,9} \right)} \right\}

10 .

\left\{ {\left( {-3{\text{ }},9} \right),\left( {-2,4} \right),\left( {-1,1} \right),\left( {0,0} \right),\left( {1,1} \right),\left( {2,4} \right),\left( {3,9} \right)} \right\}

11 .

\left\{ {\left( {9,-3} \right),\left( {4,-2} \right),\left( {1,-1} \right),\left( {0,0} \right),\left( {1,1} \right),\left( {4,{\text{ }}2} \right),\left( {9,3} \right)} \right\}

For the following exercises, determine whether the mapping represented a function.

12 .

Mapping of names and birthdays. Names and birthdays are as follows. Rebecca: January 18. Jennifer: April 1. John: January 18. Hector: June 23. Luis: February 15. Ebony: April 7. Raphael: November 6. Meredith: August 19. Karen: August 19. Joseph: July 30.

13 .

Mapping of names and birthdays. Names and birthdays are as follows. Amy: February 24. Carol: May 30. Devon: January 5. Harrison: January 7. Jackson: November 26. Labron: April 7. Mason: July 20. Natalie: March 1. Paul: August 1. Sylvester: November 13.

For the following exercises, determine whether the equations represent

y

as a function of

x

.

14 .

5x+2y=10

15 .

y = {x^2}

16 .

x = {y^2}

17 .

3{x^2} + y = 14

18 .

2x + {y^2} = 6

19 .

y = - 2{x^2} + 40x

For the following exercises, use the vertical line test to determine which graph represents a function.

20 .

A polynomial function is plotted on an x y coordinate plane. The x and y axes range from negative 5 to 5, in increments of 1. The function passes through the points, (negative 2, 0), (negative 1, 3), (0, 0), (1, negative 3), and (2, 0). Note: all values are approximate.

21 .

Two rays are plotted on an x y coordinate plane. The x and y axes range from negative 5 to 5, in increments of 1. The first ray passes through the points, (0, 0), (1, 0), (2, 1), and (3, 4). The second ray passes through the points, (0, 0), (1, 0), (2, negative 1), and (3, negative 4).

22 .

A hyperbola is plotted on an x y coordinate plane. The x and y axes range from negative 5 to 5, in increments of 1. One arm of the hyperbola passes through the points, (negative 3, 0.5), (negative 1.5, 1), and (negative 0.5, 5). The other arm passes through the points, (0.5, negative 3), (0.8, negative 1.5), and (5, negative 0.5). Note: all values are approximate.

23 .

A line is plotted on an x y coordinate plane. The x and y axes have 10 units, each. The line passes through the points, (negative 5, negative 2), (negative 1, negative 2), (0, 0), (1, 2), and (5, 2).

24 .

An ellipse is plotted on an x y coordinate plane. The x and y axes have 10 units, each. The center of the ellipse is at (2.5, negative 3). The ellipse passes through the points, (1, negative 3), (2.5, negative 2), (2.5, negative 4), and (4, negative 3).

25 .

A polynomial function is plotted on an x y coordinate plane. The x and y axes have 10 units, each. The function passes through the points, (negative 2, negative 2), (negative 1, 0), (0, 0), (1, 0), and (2, 2).

For the following exercises, use the set of ordered pairs to find the domain and the range.

26 .

\left\{ {\left( {-3,9} \right),\left( {-2,4} \right),\left( {-1,1} \right),\left( {0,0} \right),\left( {1,1} \right),\left( {2,4} \right),\left( {3,9} \right)} \right\}

27 .

\left\{ {\left( {-9,-3} \right),\left( {-4,-2} \right),\left( {-1,-1} \right),\left( {0,0} \right),\left( {1,1} \right),\left( {4,2} \right),\left( {9,3} \right)} \right\}

28 .

\left\{ {\left( {-3,27} \right),\left( {-2,8} \right),\left( {-1,1} \right),\left( {0,0} \right),\left( {1,1} \right),\left( {2,8} \right),\left( {3,27} \right)} \right\}

29 .

\left\{ {\left( {-3,-27} \right),\left( {-2,-8} \right),\left( {-1,-1} \right),\left( {0,0} \right),\left( {1,1} \right),\left( {2,8} \right),\left( {3,27} \right)} \right\}

For the following exercises, use the graph to find the domain and the range.

30 .

Six points are plotted on an x y coordinate plane. The x and y axes range from negative 5 to 5, in increments of 1. The points are plotted at the following coordinates: (negative 3, 4), (negative 2, negative 1), (0, negative 3), (2, 3), (4, negative 1), and (4, negative 3).

31 .

32 .

33 .

5.7 Functions

Learning Objectives

Use Function Notation

Evaluating the Function

Solution

Evaluating the Function in an Application

Solution

Determining If a Relation Is a Function with Different Representations

Determining If a Relation Is a Function with a Set of Ordered Pairs

Solution

Determining If a Relation Is a Function with Mapping

Solution

Determining If a Relation Is a Function with an Equation

Solution

Applying the Vertical Line Test

Applying the Vertical Line Test

Solution

Applying the Vertical Line Test to a Parabola

Solution

Determining the Domain and Range of a Function

Finding the Domain and Range of Ordered Pairs

Solution

Finding the Domain and Range on a Graph

Solution

Function and Function Notation

Check Your Understanding

Section 5.7 Exercises