1.2 Functions

Definition of Function

We often want to predict values of one variable from the values of a related variable. For example, when a physician prescribes a drug in a certain dosage, she needs to know how long the dose will remain in the bloodstream. A sales manager needs to know how the price of his product will affect its sales. A function is a special type of relationship between variables that allows us to make such predictions.

Suppose it costs $800 for flying lessons, plus $30 per hour to rent a plane. If we let $C$ represent the total cost for $t$ hours of flying lessons, then

$C = 800 + 30 t (t \geq 0)$

Thus, for example

when	$t equals 0$ ,	$C = 800 + 30 (0) = 800$
when	$t equals 4$ ,	$C = 800 + 30 (4) = 920$
when	$t equals 10$ ,	$C = 800 + 30 (10) = 1100$

The variable $t$ is called the input or independent variable, and $C$ is the output or dependent variable, because its values are determined by the value of $t$ . We can display the relationship between two variables by a table or by ordered pairs. The input variable is the first component of the ordered pair, and the output variable is the second component.

$t$	$C$	$(t, C)$
$0$	$800$	$open parenthesis 0, 800 close parenthesis$
$4$	$920$	$open parenthesis 4, 920 close parenthesis$
$10$	$1100$	$open parenthesis 10, 1100 close parenthesis$

For this relationship, we can find the value of $C$ for any given value of $t$ . All we have to do is substitute the value of $t$ into the equation and solve for $C$ . Note that there can be only one value of $C$ for each value of $t$ .

As part of a project to improve the success rate of freshmen, the counseling department studied the grades earned by a group of students in English and algebra. Do you think that a student's grade in algebra is a function of his or her grade in English? _____
Explain why or why not.
_____
Phatburger features a soda bar, where you can serve your own soft drinks in any size. Do you think that the number of calories in a serving of Zap Kola is a function of the number of fluid ounces? _____
Explain why or why not.
_____

No, students with the same grade in English can have different grades in algebra.
Yes, the number of calories is proportional to the number of fluid ounces.

As part of a project to improve the success rate of freshmen, the counseling department studied the grades earned by a group of students in English and algebra. Do you think that a student's grade in algebra is a function of his or her grade in English? Explain why or why not.
Phatburger features a soda bar, where you can serve your own soft drinks in any size. Do you think that the number of calories in a serving of Zap Kola is a function of the number of fluid ounces? Explain why or why not.

No, students with the same grade in English can have different grades in algebra.
Yes, the number of calories is proportional to the number of fluid ounces.

What distinguishes a function from other variable relationships?

_____

There cannot be two output values for a single input value.

What distinguishes a function from other variable relationships?

The variables are related by a formula.
The values of the input and output variables must be different.
There cannot be two output values for a single input value.
There cannot be two input values for a single output value.

A function can be described in several different ways. In the following examples, we consider functions defined by tables, by graphs, and by equations.

Functions Defined by Tables

When we use a table to describe a function, the first variable in the table (the left column of a vertical table or the top row of a horizontal table) is the input variable, and the second variable is the output. We say that the output variable is a function of the input.

Example 2

The table below shows data on sales compiled over several years by the accounting office for Eau Claire Auto Parts, a division of Major Motors. In this example, the year is the input variable, and total sales is the output. We say that total sales, $S$ , is a function of $t$ .
Year $(t)$ Total sales $(S)$
2000 $612,000
2001 $663,000
2002 $692,000
2003 $749,000
2004 $904,000

The table below gives the cost of sending printed material by first-class mail starting July 10, 2022.

Weight in ounces $(w)$	Postage $(P)$
$0 less than w less than or equal to 1$	$0.60
$1 less than w less than or equal to 2$	$0.84
$2 less than w less than or equal to 3$	$1.08
$3 less than w less than or equal to 4$	$1.32
$4 less than w less than or equal to 5$	$1.56
$5 less than w less than or equal to 6$	$1.80
$6 less than w less than or equal to 7$	$2.04

If we know the weight of the article being shipped, we can find the postage from the table. For instance, a catalog weighing 4.5 ounces would require $1.56 in postage. In this example,

w

is the input variable and

p

is the output variable. We say that

p

is a function of

w

The table below records the age and cholesterol count for 20 patients tested in a hospital survey.

Age	Cholesterol count	Age	Cholesterol count
53	217	$51$	$209$
48	232	53	241
55	198	49	186
56	238	$51$	$216$
$51$	$227$	57	208
52	264	52	248
53	195	50	214
47	203	56	271
48	212	53	193
50	234	48	172

According to these data, cholesterol count is not a function of age, because several patients who are the same age have different cholesterol levels. For example, three different patients are 51 years old but have cholesterol counts of 227, 209, and 216, respectively. Thus, we cannot determine a unique value of the output variable (cholesterol count) from the value of the input variable (age). Other factors besides age must influence a person’s cholesterol count.

Decide whether each table describes $y$ as a function of $x$ . Explain your choice.

$x$ $3.5$ $2.0$ $2.5$ $3.5$ $2.5$ $4.0$ $2.5$ $3.0$
$y$ $2.5$ $3.0$ $2.5$ $4.0$ $3.5$ $4.0$ $2.0$ $2.5$

Is $y$ a function of $x$ ? _____
_____
$x$ $minus 3$ $minus 2$ $minus 1$ $0$ $1$ $2$ $3$
$y$ $17$ $3$ $0$ $minus 1$ $0$ $3$ $17$

Is $y$ a function of $x$ ? _____
_____

$x$	$3.5$	$2.0$	$2.5$	$3.5$	$2.5$	$4.0$	$2.5$	$3.0$
$y$	$2.5$	$3.0$	$2.5$	$4.0$	$3.5$	$4.0$	$2.0$	$2.5$

$x$	$minus 3$	$minus 2$	$minus 1$	$0$	$1$	$2$	$3$
$y$	$17$	$3$	$0$	$minus 1$	$0$	$3$	$17$

No, for example, $x equals 3.5$ corresponds both to $y equals 2.5$ and also to $y equals 4$ .
Yes, each value of $x$ has exactly one value of $y$ associated with it.

Decide whether each table describes $y$ as a function of $x$ . Explain your choice.

$x$ $3.5$ $2.0$ $2.5$ $3.5$ $2.5$ $4.0$ $2.5$ $3.0$
$y$ $2.5$ $3.0$ $2.5$ $4.0$ $3.5$ $4.0$ $2.0$ $2.5$
$x$ $minus 3$ $minus 2$ $minus 1$ $0$ $1$ $2$ $3$
$y$ $17$ $3$ $0$ $minus 1$ $0$ $3$ $17$

$x$	$3.5$	$2.0$	$2.5$	$3.5$	$2.5$	$4.0$	$2.5$	$3.0$
$y$	$2.5$	$3.0$	$2.5$	$4.0$	$3.5$	$4.0$	$2.0$	$2.5$

$x$	$minus 3$	$minus 2$	$minus 1$	$0$	$1$	$2$	$3$
$y$	$17$	$3$	$0$	$minus 1$	$0$	$3$	$17$

No, for example, $x equals 3.5$ corresponds both to $y equals 2.5$ and also to $y equals 4$ .
Yes, each value of $x$ has exactly one value of $y$ associated with it.

How would you know if a table of values does not come from a function?

_____

Two different output values have the same input value.

How would you know if a table of values does not come from a function?

The output values are all the same.
The input values are not evely spaced.
Two different input values have the same output value.
Two different output values have the same input value.

Functions Defined by Graphs

We can also use a graph to define a function. The input variable is displayed on the horizontal axis, and the output variable on the vertical axis.

Example 3

The graph shows the number of hours, $H$ , that the sun is above the horizon in Peoria, Illinois, on day $t$ , where $t equals 0$ on January 1.

Which variable is the input, and which is the output?
How many hours of sunlight are there in Peoria on day 150?
On which days are there 12 hours of sunlight?
What are the maximum and minimum values of $H$ , and when do these values occur?

The input variable, $t$ , appears on the horizontal axis. The number of daylight hours, $H$ , is a function of the date. The output variable appears on the vertical axis.
The point on the curve where $t equals 150$ has $H approximately equals 14.1$ , so Peoria gets about 14.1 hours of daylight when $t equals 150$ , which is at the end of May.
$H equals 12$ at the two points where $t approximately equals 85$ (in late March) and $t approximately equals 270$ (late September).
The maximum value of 14.4 hours occurs on the longest day of the year, when $t approximately equals 170$ , about three weeks into June. The minimum of 9.6 hours occurs on the shortest day, when $t approximately equals 355$ , about three weeks into December.

The graph shows the elevation in feet, $a$ , of the Los Angeles Marathon course at a distance $d$ miles into the race. (Source: Los Angeles Times, March 3, 2005)

Which variable is the input, and which is the output?
_____
What is the elevation at mile 20?
Answer: _____ feet
At what distances is the elevation 150 feet?
The relevant distances (to the nearest half-mile) separated by commas: _____ miles
What are the maximum and minimum values of $a$ , and when do these values occur?
The maximum elevation is $a =$ _____ feet which occurs at $d =$ _____.
The runners pass by the Los Angeles Coliseum at about 4.2 miles into the race. What is the elevation there?
Approximately (within 5) _____ feet

The input variable is $d$ , and the output variable is $a$ .
Approximately 210 feet
Approximately where $d approximately equals 5$ , $d approximately equals 11$ , $d approximately equals 12$ , $d approximately equals 16$ , $d approximately equals 17.5$ , and $d approximately equals 18$
The maximum value of 300 feet occurs at the start, when $d equals 0$ . The minimum of 85 feet occurs when $d approximately equals 15$ .
Approximately 165 feet

The graph shows the elevation in feet, $a$ , of the Los Angeles Marathon course at a distance $d$ miles into the race. (Source: Los Angeles Times, March 3, 2005)

Which variable is the input, and which is the output?
What is the elevation at mile 20?
At what distances is the elevation 150 feet?
What are the maximum and minimum values of $a$ , and when do these values occur?
The runners pass by the Los Angeles Coliseum at about 4.2 miles into the race. What is the elevation there?

The input variable is $d$ , and the output variable is $a$ .
Approximately 210 feet
Approximately where $d approximately equals 5$ , $d approximately equals 11$ , $d approximately equals 12$ , $d approximately equals 16$ , $d approximately equals 17.5$ , and $d approximately equals 18$
The maximum value of 300 feet occurs at the start, when $d equals 0$ . The minimum of 85 feet occurs when $d approximately equals 15$ .
Approximately 165 feet

Functions Defined by Equations

Example illustrates a function defined by an equation.

Example 4

As of 2020, One World Trade Center in New York City is the nation’s tallest building, at 1776 feet. If an algebra book is dropped from the top of One World Trade Center, its height above the ground after $t$ seconds is given by the equation

$h = 1776 - 16 t^{2}$

Thus, after $1$ second the book’s height is

$h = 1776 - 16 (1)^{2} = 1760 feet$

After $2$ seconds its height is

$h = 1776 - 16 (2)^{2} = 1712 feet$

For this function, $t$ is the input variable and $h$ is the output variable. For any value of $t$ , a unique value of $h$ can be determined from the equation for $h$ . We say that $h$ is a function of $t$ .

Write an equation that gives the volume, $V$ , of a sphere as a function of its radius, $r$ .

$V =$ _____

Note: Use "pi" to enter the number $π$ .

$V = \frac{4}{3} π r^{3}$

Write an equation that gives the volume, $V$ , of a sphere as a function of its radius, $r$ .

$V = \frac{4}{3} π r^{3}$

Name three ways to describe a function.

_____

By tables, equations, or graphs

Name three ways to describe a function.

By inputs, outputs, or evaluation
By tables, equations, or graphs
By the intercepts, the slope, or the vertex
By numbers, letters, or diagrams

Technology

Making a Table of Values with a Calculator

We can use a graphing calculator to make a table of values for a function defined by an equation. For the function in Example,

$h = 1776 - 16 t^{2}$

we follow the steps:

Enter the equation: Press the Y= key, clear out any other equations, and define $Y_{1} = 1776 - 16 X^{2} .$
Choose the $x$ -values for the table. Press 2ndWINDOW to access the $T b l S e t$ (Table Setup) menu and set it to look like the figure at left below.

This setting will give us an initial x-value of 0 $open parenthesis T b l S t a r t equals 0 close parenthesis$ and an increment of one unit in the $x$ -values, $open parenthesis Δ T b l equals 1 close parenthesis$ . It also fills in values of both variables automatically.

Press 2nd GRAPH to see the table of values, as shown in the figure at right below. From this table, we can check the heights we found in Example.

Now try making a table of values with $T b l S t a r t equals 0$ and $Δ T b l equals 0.5$ . Use the and arrow keys to scroll up and down the table.

Write one question you still have about functions.

_____

Write one question you still have about functions.

Function Notation

There is a convenient notation for discussing functions. First, we choose a letter, such as $f$ , $g$ , or $h$ (or $F$ , $G$ , or $H$ ), to name a particular function. (We can use any letter, but these are the most common choices.)

For instance, in Example, the height, $h$ , of a falling algebra book is a function of the elapsed time, $t$ . We might call this function $f$ . In other words, $f$ is the name of the relationship between the variables $h$ and $t$ . We write

$h = f (t)$

which means " $h$ is a function of $t$ , and $f$ is the name of the function."

With this new notation we may write

$h = f (t) = 1776 - 16 t^{2}$

or just

$f (t) = 1776 - 16 t^{2}$

instead of

$h = 1776 - 16 t^{2}$

to describe the function.

Example 5

In Example, the height of an algebra book dropped from the top of One World Trade Center is given by the equation

$h = 1776 - 16 t^{2}$

We see that

when $t equals 1$		$h equals 1760$
when $t equals 2$		$h equals 1712$

Using function notation, these relationships can be expressed more concisely as

f open parenthesis 1 close parenthesis equals 1760

and

f open parenthesis 2 close parenthesis equals 1712

which we read as " $f$ of $1$ equals $1760$ " and " $f$ of $2$ equals $1712$ ." The values for the input variable, $t$ , appear inside the parentheses, and the values for the output variable, $h$ , appear on the other side of the equation.

Remember that when we write $y = f (x)$ , the symbol $f (x)$ is just another name for the output variable.

True or False.

The notation $f (t)$ indicates the product of $f$ and $t$ . _____
If $y = f (x)$ , then $f (x)$ gives the value of the input variable. _____
If $Q$ is a function of $M$ , we may write $M = f (Q)$ . _____
In the equation $d = g (n)$ the letters $d, g,$ and $n$ are variables. _____

False
False
False
False

Decide whether each statement is true or false.

The notation $f (t)$ indicates the product of $f$ and $t$ .
If $y = f (x)$ , then $f (x)$ gives the value of the input variable.
If $Q$ is a function of $M$ , we may write $M = f (Q)$ .
In the equation $d = g (n)$ the letters $d, g,$ and $n$ are variables.

Let $F$ be the name of the function defined by the graph in Example, the number of hours of daylight in Peoria $t$ days after January 1.

Use function notation to state that $H$ is a function of $t$ .
_____
What does the statement $F open parenthesis 15 close parenthesis equals 9.7$ mean in the context of the problem?
_____

$H = F (t)$
The sun is above the horizon in Peoria for 9.7 hours on January 16.

Let $F$ be the name of the function defined by the graph in Example, the number of hours of daylight in Peoria $t$ days after January 1.

Use function notation to state that $H$ is a function of $t$ .
What does the statement $F open parenthesis 15 close parenthesis equals 9.7$ mean in the context of the problem?

$H = F (t)$
The sun is above the horizon in Peoria for 9.7 hours on January 16.

Use function notation to write the statement " $L$ defines $w$ as a function of $p$ ."

_____

$w = L (p)$

Use function notation to write the statement " $L$ defines $w$ as a function of $p$ ."

$L = w (p)$
$w = L (p)$
$p = L (w)$
$L = p (w)$

Using Function Notation

Finding the value of the output variable that corresponds to a particular value of the input variable is called evaluating the function.

Example 6

Let $g$ be the name of the postage function defined by the table in Example b. Find $g open parenthesis 1 close parenthesis$ , $g open parenthesis 3 close parenthesis$ , and $g open parenthesis 6.75$ ).

According to the table,

when $w equals 1$ ,		$p equals 0.60$	so	$g open parenthesis 1 close parenthesis equals 0.60$
when $w equals 3$ ,		$p equals 1.08$	so	$g open parenthesis 3 close parenthesis equals 1.08$
when $w equals 6.75$ ,		$p equals 2.04$	so	$g open parenthesis 6.75 close parenthesis equals 2.04$

Thus, a letter weighing 1 ounce costs $0.60 to mail, a letter weighing 3 ounces costs $1.08, and a letter weighing 6.75 ounces costs $2.04.

When you exercise, your heart rate should increase until it reaches your target heart rate. The table shows target heart rate, $r = f (a)$ , as a function of age.

$a$	20	25	30	35	40	45	50	55	60	65	70
$r$	150	146	142	139	135	131	127	124	120	116	112

Find $f open parenthesis 25 close parenthesis$ and $f open parenthesis 50 close parenthesis$ .
$f open parenthesis 25 close parenthesis equals$ _____
$f open parenthesis 50 close parenthesis equals$ _____
Find a value of $a$ for which $f open parenthesis a close parenthesis equals 135$ .
$a =$ _____

$f open parenthesis 25 close parenthesis equals 146$ , $f open parenthesis 50 close parenthesis equals 127$
$a equals 40$

When you exercise, your heart rate should increase until it reaches your target heart rate. The table shows target heart rate, $r = f (a)$ , as a function of age.

$a$	20	25	30	35	40	45	50	55	60	65	70
$r$	150	146	142	139	135	131	127	124	120	116	112

Find $f open parenthesis 25 close parenthesis$ and $f open parenthesis 50 close parenthesis$ .
Find a value of $a$ for which $f open parenthesis a close parenthesis equals 135$ .

$f open parenthesis 25 close parenthesis equals 146$ , $f open parenthesis 50 close parenthesis equals 127$
$a equals 40$

If $n = f (a)$ , what are the input and output variables?

_____

$a$ is the input and $n$ is the output

If $n = f (a)$ , what are the input and output variables?

$f$ is the output and $n$ is the input.
$a$ is the output and $f$ is the input.
$a$ is the input and $n$ is the output.
$f (a)$ is the output and $n$ is the input.

If a function is described by an equation, we simply substitute the given input value into the equation to find the corresponding output, or function value.

Example 7

The function $H$ is defined by $H = f (s) = \frac{\sqrt{s + 3}}{s}$ . Evaluate the function at the following values.

$s equals 6$
$s = - 1$

$f (6) = \frac{\sqrt{6 + 3}}{6} = \frac{\sqrt{9}}{6} = \frac{3}{6} = \frac{1}{2}$ . Thus, $f (6) = \frac{1}{2}$ .
$f (- 1) = \frac{\sqrt{- 1 + 3}}{- 1} = \frac{\sqrt{2}}{- 1} = - \sqrt{2}$ . Thus, $f (- 1) = - \sqrt{2}$ .

Complete the table displaying ordered pairs for the function $f (x) = 5 - x^{3}$ . Evaluate the function to find the corresponding $f (x)$ -value for each value of $x$ .

$x$	$f (x)$
$minus 2$	_____	$f (- 2) = 5 - (- 2)^{3} =$
$0$	_____	$f (0) = 5 - 0^{3} =$
$1$	_____	$f (1) = 5 - 1^{3} =$
$3$	_____	$f (3) = 5 - 3^{3} =$

$x$	$f (x)$
$minus 2$	$13$
$0$	$5$
$1$	$4$
$3$	$minus 22$

Complete the table displaying ordered pairs for the function $f (x) = 5 - x^{3}$ . Evaluate the function to find the corresponding $f (x)$ -value for each value of $x$ .

$x$	$f (x)$
$minus 2$		$f (- 2) = 5 - (- 2)^{3} =$
$0$		$f (0) = 5 - 0^{3} =$
$1$		$f (1) = 5 - 1^{3} =$
$3$		$f (3) = 5 - 3^{3} =$

$x$	$f (x)$
$minus 2$	$13$
$0$	$5$
$1$	$4$
$3$	$minus 22$

Technology

Evaluating a Function

We can use the table feature on a graphing calculator to evaluate functions. Consider the function of Practice 7, $f (x) = 5 - x^{3}$ .

Press Y=, clear any old functions, and enter
$Y_{1} = 5 - X$ ^ $3$
Press TblSet (2nd WINDOW) and choose $A s k$ after $I n d p n t$ , as shown in the figure at left below, and press ENTER. This setting allows you to enter any $x$ -values you like.
Press TABLE (using 2nd GRAPH).
To follow Practice 7, key in (-) 2 ENTER for the $x$ -value, and the calculator will fill in the $y$ -value. Continue by entering 0, 1, 3, or any other $x$ -values you choose.
To follow Practice 7, key in (-) 2 ENTER for the $x$ -value, and the calculator will fill in the $y$ -value. Continue by entering 0, 1, 3, or any other $x$ -values you choose.

One such table is shown in the figure at right below.

If you would like to evaluate a new function, you do not have to return to the Y= screen. Use the and arrow keys to highlight $Y sub 1$ at the top of the second column. The definition of $Y sub 1$ will appear at the bottom of the display, as shown above. You can key in a new definition here, and the second column will be updated automatically to show the $y$ -values of the new function.

To simplify the notation, we sometimes use the same letter for the output variable and for the name of the function. In the next example, $C$ is used in this way.

The volume of a sphere of radius $r$ centimeters is given by

$V = V (r) = \frac{4}{3} π r^{3}$

Evaluate $V open parenthesis 10 close parenthesis$ and explain what it means.

Note: You may use "pi" to enter the number $π$ .

$V open parenthesis 10 close parenthesis equals$ _____, which represents

_____

$V (10) = 4000 π / 3 \approx 4188.79 {cm}^{3}$ is the volume of a sphere whose radius is 10 cm.

The volume of a sphere of radius $r$ centimeters is given by

$V = V (r) = \frac{4}{3} π r^{3}$

Evaluate $V open parenthesis 10 close parenthesis$ and explain what it means.

$V (10) = 4000 π / 3 \approx 4188.79 {cm}^{3}$ is the volume of a sphere whose radius is 10 cm.

Do linear equations $y = m x + b$ and quadratic equations $y = a x^{2} + b x + c$ define functions? Why or why not?

_____

Do linear equations $y = m x + b$ and quadratic equations $y = a x^{2} + b x + c$ define functions? Why or why not?

Operations with Function Notation

Sometimes we need to evaluate a function at an algebraic expression rather than at a specific number.

Example 9

TrailGear manufactures backpacks at a cost of

$C (x) = 3000 + 20 x$

for $x$ backpacks. The company finds that the monthly demand for backpacks increases by 50% during the summer. The backpacks are produced at several small co-ops in different states.

If each co-op usually produces $b$ backpacks per month, how many should it produce during the summer months?
What costs for producing backpacks should the company expect during the summer?

An increase of 50% means an additional 50% of the current production level, $b$ . Therefore, a co-op that produced $b$ backpacks per month during the winter should increase production to $b plus 0.5 b$ , or $1.5 b$ backpacks per month in the summer.
The cost of producing $1.5 b$ backpacks will be
$C (1.5 b) = 3000 + 20 (1.5 b) = 3000 + 30 b$

A spherical balloon has a radius of 10 centimeters.

If we increase the radius by $h$ centimeters, what will the new volume be?
_____
If $h equals 2$ , how much did the volume increase? Round your answer to hundredths.
It increased by _____ ${cm}^{3}$ .

Note: Use "pi" to enter the number $π$ .

$V (10 + h) = \frac{4}{3} π (10 + h)^{3} {cm}^{3}$
From $V open parenthesis 10 close parenthesis$ to $V open parenthesis 12 close parenthesis$ is an increase of about $3049.44 cm cubed$

A spherical balloon has a radius of 10 centimeters.

If we increase the radius by $h$ centimeters, what will the new volume be?
If $h equals 2$ , how much did the volume increase? Round your answer to hundredths.

$V (10 + h) = \frac{4}{3} π (10 + h)^{3} {cm}^{3}$
From $V open parenthesis 10 close parenthesis$ to $V open parenthesis 12 close parenthesis$ is an increase of about $3049.44 cm cubed$

Example 10

Evaluate the function $f (x) = 4 x^{2} - x + 5$ for the following expressions.

$x equals 2 h$
$x = a + 3$

$\begin{aligned} f (2 h) & = 4 (2 h)^{2} - (2 h) + 5 \\ = 4 (4 h^{2}) - 2 h + 5 \\ = 16 h^{2} - 2 h + 5 \end{aligned}$
$\begin{aligned} f (a + 3) & = 4 (a + 3)^{2} - (a + 3) + 5 \\ = 4 (a^{2} + 6 a + 9) - a - 3 + 5 \\ = 4 a^{2} + 24 a + 36 - a + 2 \\ = 4 a^{2} + 23 a + 38 \end{aligned}$

Caution

In Example, notice that

$f open parenthesis 2 h close parenthesis not equal to 2 f open parenthesis h close parenthesis$

and

$f (a + 3) \neq f (a) + f (3)$

To compute $f open parenthesis a close parenthesis plus f open parenthesis 3 close parenthesis$ , we must first compute $f (a)$ and $f open parenthesis 3 close parenthesis$ , then add them:

$\begin{aligned} f (a) + f (3) & = (4 a^{2} - a + 5) + (4 \cdot 3^{2} - 3 + 5) \\ = 4 a^{2} - a + 43 \end{aligned}$

In general, it is not true that $f (a + b) = f (a) + f (b)$ . Remember that the parentheses in the expression $f (x)$ do not indicate multiplication, so the distributive law does not apply to the expression $f (a + b)$ .

Define the function $f (x) = x^{2}$ . Which of these is equal to $f (x + y)$ ?

_____

$(x + y)^{2}$

Define the function $f (x) = x^{2}$ . Which of these is equal to $f (x + y)$ ?

$f (x) + f (y)$
$f (x + y)^{2}$
$x^{2} + y^{2}$
$(x + y)^{2}$

Let $f (x) = x^{3} - 1$ and evaluate each expression.

$f (2) + f (3) =$ _____
$f (2 + 3) =$ _____
$2 f open parenthesis x close parenthesis plus 3$ _____

$33$
$124$
$2 x cubed plus 1$

Let $f (x) = x^{3} - 1$ and evaluate each expression.

$f open parenthesis 2 close parenthesis plus f open parenthesis 3 close parenthesis$
$f open parenthesis 2 plus 3 close parenthesis$
$2 f open parenthesis x close parenthesis plus 3$

$33$
$124$
$2 x cubed plus 1$

Explain why $f (a + b)$ is not the same as $f (a) + f (b)$ for most functions.

_____

Explain why $f (a + b)$ is not the same as $f (a) + f (b)$ for most functions.

Section Summary

Vocabulary

Look up the definitions of new terms in the Glossary.

Function
Input variable
Independent variable
Function value
Dependent variable
Output variable

CONCEPTS

A function is a rule that assigns to each value of the input variable a unique value of the output variable.
Functions may be defined by words, tables, graphs, or equations.
Function notation: $y = f (x)$ , where $x$ is the input and $y$ is the output.

STUDY QUESTIONS

What property makes a relation between two variables a function?
Name three ways to define a function.
Give an example of a function in which two distinct values of the input variable correspond to the same value of the output variable.
Use function notation to write the statement " $G$ defines $w$ as a function of $p$ ."
Give an example of a function for which $f (2 + 3) \neq f (2) + f (3)$ .

SKILLS

Practice each skill in the Homework problems listed.

Decide whether a relationship between two variables is a function: #1–26
Evaluate a function defined by a table, a graph, or an equation: #27–54
Choose appropriate scales for the axes: #5–12
Interpret function notation: #31–34, 49–54
Simplify expressions involving function notation: #59–76

Homework 1.2

For which of Problems 1–6 is the second quantity a function of the first? Explain your answers.

Price of an item; sales tax on the item at 4%

Function; the tax is determined by the price of the item.

Time traveled at constant speed; distance traveled

Number of years of education; annual income

Not a function; incomes may differ for same number of years of education.

Distance flown in an airplane; price of the ticket

Volume of a container of water; the weight of the water

Function; weight is determined by volume.

Amount of a paycheck; amount of Social Security tax withheld

Each of the objects in Problems 7–14 establishes a correspondence between two variables. Suggest appropriate input and output variables and decide whether the relationship is a function.

An itemized grocery receipt

Input: items purchased; output: price of item. Yes, a function because each item has only one price.

An inventory list

An index

Input: topics; output: page or pages on which topic occurs. No, not a function because the same topic may appear in more than one page.

A will

An instructor's grade book

Input: students’ names; output: students’ scores on quizzes, tests, etc. No, not a function because the same student can have different grades on different tests.

An address book

A bathroom scale

Input: person stepping on scales; output: person's weight. Yes, a function because a person cannot have two different weights at the same time.

A radio dial

Which of the tables in Problems 15–26 define the second variable as a function of the first variable? Explain why or why not.

$x$	$t$
$minus 1$	$2$
$0$	$9$
$1$	$minus 2$
$0$	$minus 3$
$minus 1$	$5$

$y$	$w$
$0$	$8$
$1$	$12$
$3$	$7$
$5$	$minus 3$
$7$	$4$

$x$	$y$
$minus 3$	$8$
$minus 2$	$3$
$minus 1$	$0$
$0$	$minus 1$
$1$	$0$
$2$	$3$
$3$	$8$

Yes

$s$	$t$
$2$	$5$
$4$	$10$
$6$	$15$
$8$	$20$
$6$	$25$
$4$	$30$
$2$	$35$

$r$	$minus 4$	$minus 2$	$0$	$2$	$4$
$v$	$6$	$6$	$3$	$6$	$8$

Yes

$p$	$minus 5$	$minus 4$	$minus 3$	$minus 2$	$minus 1$
$d$	$minus 5$	$minus 4$	$minus 3$	$minus 2$	$minus 1$

Pressure ( $p$ )	Volume ( $v$ )
$15$	$100.0$
$20$	$75.0$
$25$	$60.0$
$30$	$50.0$
$35$	$42.8$
$40$	$37.5$
$45$	$33.3$
$50$	$30.0$

Yes

Frequency ( $f$ )	Wavelength ( $w$ )
$5$	$60.0$
$10$	$30.0$
$20$	$15.0$
$30$	$10.0$
$40$	$7.5$
$50$	$6.0$
$60$	$5.0$
$70$	$4.3$

Temperature ( $T$ )	Humidity ( $h$ )
Jan. 1 $34 degrees$ F	$42 %$
Jan. 2 $36 degrees$ F	$44 %$
Jan. 3 $35 degrees$ F	$47 %$
Jan. 4 $29 degrees$ F	$50 %$
Jan. 5 $31 degrees$ F	$52 %$
Jan. 6 $35 degrees$ F	$51 %$
Jan. 7 $34 degrees$ F	$49 %$

Inflation rate ( $I$ )	Unemployment rate ( $U$ )
1972 $5.6 %$	$5.1 %$
1973 $6.2 %$	$4.5 %$
1974 $10.1 %$	$4.9 %$
1975 $9.2 %$	$7.4 %$
1976 $5.8 %$	$6.7 %$
1977 $5.6 %$	$6.8 %$
1978 $6.7 %$	$7.4 %$

Adjusted gross income ( $I$ )	Tax bracket ( $T$ )
$ $0 minus 2479$	$0 %$
$ $2480 minus 3669$	$4.5 %$
$ $3670 minus 4749$	$12 %$
$ $4750 minus 7009$	$14 %$
$ $7010 minus 9169$	$15 %$
$ $9170 minus 11, 649$	$16 %$
$ $11, 650 minus 13, 919$	$18 %$

Yes

Cost of merchandise ( $M$ )	Shipping charge ( $C$ )
$$ 0.01 minus 10.00$	$$ 2.50$
$10.01 minus 20.00$	$3.75$
$20.01 minus 35.00$	$4.85$
$35.01 minus 50.00$	$5.95$
$50.01 minus 75.00$	$6.95$
$75.01 minus 100.00$	$7.95$
Over $100.00$	$8.95$

The function described in Problem 21 is called $g$ , so that $v = g (p)$ . Find the following:

$g open parenthesis 25 close parenthesis$
$g open parenthesis 40 close parenthesis$
$x$ so that $g open parenthesis x close parenthesis equals 50$

$60$
$37.5$
$30$

The function described in Problem 22 is called $h$ , so that $w = h (f)$ . Find the following:

$h open parenthesis 20 close parenthesis$
$h open parenthesis 60 close parenthesis$
$x$ so that $h open parenthesis x close parenthesis equals 10$

The function described in Problem 25 is called $T$ , so that $T = T (I)$ . Find the following:

$T open parenthesis 8750 close parenthesis$
$T open parenthesis 6249 close parenthesis$
$x$ so that $T open parenthesis x close parenthesis equals 15 %$

$15 %$
$14 %$
$7010–$9169

The function described in Problem 26 is called $C$ , so that $C = C (M)$ . Find the following:

$C open parenthesis 11.50 close parenthesis$
$C open parenthesis 47.24 close parenthesis$
$x$ so that $C open parenthesis x close parenthesis equals 7.95$

Data indicate that U.S. women are delaying having children longer than their counterparts 50 years ago. The table shows $f (t)$ the percent of 20–24-year-old women in year $t$ who had not yet had children. (Source: U.S. Dept of Health and Human Services)

Year ( $t$ )	$1960$	$1965$	$1970$	$1975$	$1980$	$1985$	$1990$	$1995$	$2000$
Percent of women	$47.5$	$51.4$	$47.0$	$62.5$	$66.2$	$67.7$	$68.3$	$65.5$	$66.0$

Evaluate $f open parenthesis 1985 close parenthesis$ and explain what it means.
Estimate a solution to the equation $f open parenthesis t close parenthesis equals 68$ and explain what it means.
In 1997, $64.9 %$ of 20–24-year-old women had not yet had children. Write an equation with function notation that states this fact.

$67.7$ : In 1985, $67.7 %$ of 20–24 year old women had not yet had children.
1987: Approximately $68 %$ of 20–24 year old women had not yet had children in 1987.
$f open parenthesis 1997 close parenthesis equals 64.9$

The table shows $f (t)$ , the death rate (per 100,000 people) from HIV among 15–24-year-olds, and $g (t)$ , the death rate from HIV among 25–34-year-olds, for selected years from 1997 to 2002. (Source: U.S. Dept of Health and Human Services)

Year	$1987$	$1988$	$1989$	$1990$	$1992$	$1994$	$1996$	$1998$	$2000$	$2002$
15–24-year-olds	$1.3$	$1.4$	$1.6$	$1.5$	$1.6$	$1.8$	$1.1$	$0.6$	$0.5$	$0.4$
25–34-year-olds	$11.7$	$14.0$	$17.9$	$19.7$	$24.2$	$28.6$	$19.2$	$8.1$	$6.1$	$4.6$

Evaluate $f open parenthesis 1996 close parenthesis$ and explain what it means.
Find a solution to the equation $g open parenthesis t close parenthesis equals 28.6$ and explain what it means.
In 1988, the death rate from HIV for 25–34-year-olds was $10$ times the corresponding rate for 15–24-year-olds. Write an equation with function notation that states this fact.

When you exercise, your heart rate should increase until it reaches your target heart rate. The table shows target heart rate, $r = f (a)$ , as a function of age.

$a$	$20$	$25$	$30$	$35$	$40$	$45$	$50$	$55$	$60$	$65$	$70$
$r$	$150$	$146$	$142$	$139$	$135$	$131$	$127$	$124$	$120$	$116$	$112$

Does $f open parenthesis 50 close parenthesis equals 2 f open parenthesis 25 close parenthesis$ ?
Find a value of a for which $f open parenthesis a close parenthesis equals 2 a$ . Is $f open parenthesis a close parenthesis equals 2 a$ for all values of $a$ ?
Is $r = f (a)$ an increasing function or a decreasing function?

No
60; no
Decreasing

The table shows $M = f (d)$ , the men's Olympic record time, and $W = g (d)$ , the women's Olympic record time, as a function of the length, $d$ , of the race. For example, the women’s record in the 100 meters is 10.61 seconds, and the men’s record in the 800 meters is 1 minute, 40.91 seconds. (Source: en.wikipedia.org as of 11/11/2025)

Distance (meters)	$100$	$200$	$400$	$800$	$1500$	$5000$	$10, 000$
Men	$9.63$	$19.30$	$43.03$	$1: 40.91$	$3: 27.65$	$12: 57.82$	$26: 43.14$
Women	$10.61$	$21.34$	$48.17$	$1: 53.43$	$3: 51.29$	$14: 26.17$	$29: 17.45$

Does $f open parenthesis 800 close parenthesis equals 2 f open parenthesis 400 close parenthesis$ ? Does $g open parenthesis 400 close parenthesis equals 2 g open parenthesis 200 close parenthesis$ ?
Find a value of $d$ for which $f open parenthesis 2 d close parenthesis less than 2 f open parenthesis d close parenthesis$ . Is there a value of $d$ for which $g open parenthesis 2 d close parenthesis less than 2 g open parenthesis d close parenthesis$ ?

In Problems 35–40, use the graph of the function to answer the questions.

The graph shows $C$ as a function of $t$ . $C$ stands for the number of students (in thousands) at State University who consider themselves computer literate, and $t$ represents time, measured in years since 1990.

When did $2000$ students consider themselves computer literate?
How long did it take that number to double?
How long did it take for the number to double again?
How many students became computer literate between January 1992 and June 1993?

1991
1 yr
1 yr
About 7300

The graph shows $P$ as a function of $t$ . $P$ is the number of people in Cedar Grove who owned a portable DVD player $t$ years after 2000.

When did 3500 people own portable DVD players?
How many people owned portable DVD players in 2005?
The number of owners of portable DVD players in Cedar Grove seems to be leveling off at what number?
How many people acquired portable DVD players between 2001 and 2004?

The graph shows the revenue, $R$ , a movie theater collects as a function of the price, $d$ , it charges for a ticket.

What is the revenue if the theater charges $ $12.00$ for a ticket?
What should the theater charge for a ticket in order to collect $ $1500$ in revenue?
For what values of $d$ is $R greater than 1875$ ?

Approximately $ $1920$
$ $5$ or $ $15$
$7.50 less than d less than 12.50$

The graph shows $S$ as a function of $w$ . $S$ represents the weekly sales of a best-selling book, in thousands of dollars, $w$ weeks after it is released.

In which weeks were sales over $ $7000$ ?
In which week did sales fall below $ $5000$ on their way down?
For what values of $w$ is $S greater than 4.4$ ?

The graph shows the federal minimum wage, $M$ , as a function of time, $t$ , adjusted for inflation to reflect its buying power in 2004 dollars. (Source: www.infoplease.com)

When did the minimum wage reach its highest buying power, and what was it worth in 2004 dollars?
When did the minimum wage fall to its lowest buying power after its peak, and what was its worth at that time?
Give two years in which the minimum wage was worth $ $8$ in 2004 dollars.

1968, about $ $8.70$
1989, about $ $5.10$
1967, approximately 1970

The graph shows the U.S. unemployment rate, $U$ , as a function of time, $t$ , for the years 1985–2004. (Source: U.S. Bureau of Labor Statistics)

When did the unemployment rate reach its highest value, and what was its highest value?
When did the unemployment rate fall to its lowest value, and what was its lowest value?
Give two years in which the unemployment rate was $4.5 %$ .

In Problems 41–48, evaluate each function for the given values.

$f (x) = 6 - 2 x$

$f open parenthesis 3 close parenthesis$
$f open parenthesis minus 2 close parenthesis$
$f open parenthesis 12.7 close parenthesis$
$f (\frac{2}{3})$

$0$
$10$
$minus 19.4$
$\frac{14}{3}$

$g (t) = 5 t - 3$

$g open parenthesis 1 close parenthesis$
$g open parenthesis minus 4 close parenthesis$
$g open parenthesis 14.1 close parenthesis$
$g (\frac{3}{4})$

$h (v) = 2 v^{2} - 3 v + 1$

$h open parenthesis 0 close parenthesis$
$h open parenthesis minus 1 close parenthesis$
$h (\frac{1}{4})$
$h open parenthesis minus 6.2 close parenthesis$

$1$
$6$
$\frac{3}{8}$
$96.48$

$r (s) = 2 s - s^{2}$

$r open parenthesis 2 close parenthesis$
$r open parenthesis minus 4 close parenthesis$
$r (\frac{1}{3})$
$r open parenthesis minus 1.3 close parenthesis$

$H (z) = \frac{2 z - 3}{z + 2}$

$H open parenthesis 4 close parenthesis$
$H open parenthesis minus 3 close parenthesis$
$H (\frac{4}{3})$
$H open parenthesis 4.5 close parenthesis$

$\frac{5}{6}$
$9$
$\frac{- 1}{10}$
$\frac{12}{13} \approx 0.923$

$F (x) = \frac{1 - x}{2 x - 3}$

$F open parenthesis 0 close parenthesis$
$F open parenthesis minus 3 close parenthesis$
$F (\frac{5}{2})$
$F open parenthesis 9.8 close parenthesis$

$E (t) = \sqrt{t - 4}$

$E open parenthesis 16 close parenthesis$
$E open parenthesis 4 close parenthesis$
$E open parenthesis 7 close parenthesis$
$E open parenthesis 4.2 close parenthesis$

$the square root of 12$
$0$
$the square root of 3$
$\sqrt{0.2} \approx 0.447$

$D (r) = \sqrt{5 - r}$

$D open parenthesis 4 close parenthesis$
$D open parenthesis minus 3 close parenthesis$
$D open parenthesis minus 9 close parenthesis$
$D open parenthesis 4.6 close parenthesis$

A sport utility vehicle costs $ $28, 000$ and depreciates according to the formula

$V (t) = 28, 000 (1 - 0.08 t)$

where $V$ is the value of the vehicle after $t$ years.

Evaluate $V open parenthesis 12 close parenthesis$ and explain what it means.
Solve the equation $V open parenthesis t close parenthesis equals 0$ and explain what it means.
If this year is $t = n$ , what does $V open parenthesis n plus 2 close parenthesis$ mean?

$V open parenthesis 12 close parenthesis equals 1120$ : After 12 years, the SUV is worth $ $1120$ .
$t equals 12.5$ : The SUV has zero value after $12 \frac{1}{2}$ years.
The value 2 years later

In a profit-sharing plan, an employee receives a salary of

$S (x) = 20, 000 + 0.01 x$

where $x$ represents the company's profit for the year.

Evaluate $S open parenthesis 850, 000 close parenthesis$ and explain what it means.
Solve the equation $S open parenthesis x close parenthesis equals 30, 000$ and explain what it means.
If the company made a profit of $p$ dollars this year, what does $S open parenthesis 2 p close parenthesis$ mean?

The number of compact cars that a large dealership can sell at price $p$ is given by

$N (p) = \frac{12, 000, 000}{p}$

Evaluate $N open parenthesis 6000 close parenthesis$ and explain what it means.
As $p$ increases, does $N (p)$ increase or decrease? Why is this reasonable?
If the current price for a compact car is $D$ , what does $2 N open parenthesis D close parenthesis$ mean?

$N open parenthesis 6000 close parenthesis equals 2000$ : $2000$ cars will be sold at a price of $ $6000$ .
$N (p)$ decreases with increasing $p$ because fewer cars will be sold when the price increases.
$2 N open parenthesis D close parenthesis$ represents twice the number of cars that can be sold at the current price.

A department store finds that the market value of its Christmas-related merchandise is given by

$M (t) = \frac{600, 000}{t}, t \leq 30$

where $t$ is the number of weeks after Christmas.

Evaluate $M open parenthesis 2 close parenthesis$ and explain what it means.
As $t$ increases, does $M (t)$ increase or decrease? Why is this reasonable?
If this week is $t = n$ , what does $M open parenthesis n plus 1 close parenthesis$ mean?

The velocity of a car that brakes suddenly can be determined from the length of its skid marks, $d$ , by

$v (d) = \sqrt{12 d}$

where $d$ is in feet and $v$ is in miles per hour.

Evaluate $v open parenthesis 250 close parenthesis$ and explain what it means.
Estimate the length of the skid marks left by a car traveling at $100$ miles per hour.
Write your answer to part (b) with function notation.

$v open parenthesis 250 close parenthesis equals 54.8$ is the speed of a car that left $250$ -foot skid marks.
$833 \frac{1}{3}$ feet
$v (833 \frac{1}{3}) = 100$

The distance, $d$ , in miles that a person can see on a clear day from a height, $h$ , in feet is given by

$d (h) = 1.22 \sqrt{h}$

Evaluate $d open parenthesis 20, 320 close parenthesis$ and explain what it means.
Estimate the height you need in order to see $100$ miles.
Write your answer to part (b) with function notation.

The figure gives data about snowfall, air temperature, and number of avalanches on the Mikka glacier in Sarek, Lapland, in 1957. (Source: Leopold, Wolman, Miller, 1992)

During June and July, avalanches occurred over three separate time intervals. What were they?
Over what three time intervals did snow fall?
When was the temperature above freezing ( $0 degrees$ C)?
Using your answers to parts (a)–(c), make a conjecture about the conditions that encourage avalanches.

June 21–24, June 29–July 3, July 8–14
June 17–21, June 25–29, July 4–7
June 22–24, June 27, June 29–July 4, July 8–14
Avalanches occur when temperatures rise above freezing immediately after snowfall.

The bar graph shows the percent of Earth's surface that lies at various altitudes or depths below the surface of the oceans. (Depths are given as negative altitudes.) (Source: Open University)

Read the graph and complete the table.

Altitude (km)	Percent of Earth's surface
$minus 7$ to $minus 6$
$minus 6$ to $minus 5$
$minus 5$ to $minus 4$
$minus 4$ to $minus 3$
$minus 3$ to $minus 2$
$minus 2$ to $minus 1$
$minus 1$ to $0$
$0$ to $1$
$1$ to $2$
$2$ to $3$
$3$ to $4$
$4$ to $5$

What is the most common altitude? What is the second most common altitude??
Approximately what percent of the Earth's surface is below sea level?
The height of Mt. Everest is $8.85$ kilometers. Can you think of a reason why it is not included in the graph?

The graph shows the temperature of the ocean at various depths. (Source: Open University)

Is depth a function of temperature?
Is temperature a function of depth?
The axes are scaled in an unusual way. Why is it useful to present the graph in this way?

No
Yes
Moving downwards on the graph corresponds to moving downwards in the ocean.

The graph shows the relationship between annual precipitation, $p$ , in a region and the amount of erosion, measured in tons per square mile, $s$ . (Source: Leopold, Wolman, Miller, 1992)

Is the amount of erosion a function of the amount of precipitation?
At what annual precipitation is erosion at a maximum, and what is that maximum?
Over what interval of annual precipitation does erosion decrease?
An increase in vegetation inhibits erosion, and precipitation encourages vegetation. What happens to the amount of erosion as precipitation increases in each of these three environments?
desert shrub: $0 less than p less than 12$
grassland: $12 less than p less than 30$
forest: $30 less than p less than 60$

desert shrub:	$0 less than p less than 12$
grassland:	$12 less than p less than 30$
forest:	$30 less than p less than 60$

In Problems 59–64, evaluate the function and simplify.

$G (s) = 3 s^{2} - 6 s$

$G open parenthesis 3 a close parenthesis$
$G open parenthesis a plus 2 close parenthesis$
$G open parenthesis a close parenthesis plus 2$
$G (- a)$

$27 a squared minus 18 a$
$3 a squared plus 6 a$
$3 a^{2} - 6 a + 2$
$3 a squared plus 6 a$

$h (x) = 2 x^{2} + 6 x - 3$

$h open parenthesis 2 a close parenthesis$
$h open parenthesis a plus 3 close parenthesis$
$h open parenthesis a close parenthesis plus 3$
$h (- a)$

$g open parenthesis x close parenthesis equals 8$

$g open parenthesis 2 close parenthesis$
$g open parenthesis 8 close parenthesis$
$g open parenthesis a plus 1 close parenthesis$
$g (- x)$

$f (t) = - 3$

$f open parenthesis 4 close parenthesis$
$f open parenthesis minus 3 close parenthesis$
$f open parenthesis b minus 2 close parenthesis$
$f (- t)$

$P (x) = x^{3} - 1$

$P open parenthesis 2 x close parenthesis$
$2 P open parenthesis x close parenthesis$
$P (x^{2})$
$[ P open parenthesis x close parenthesis ] squared$

$8 x cubed minus 1$
$2 x cubed minus 2$
$x^{6} - 1$
$x^{6} - 2 x^{3} + 1$

$Q open parenthesis t close parenthesis equals 5 t cubed$

$Q open parenthesis 2 t close parenthesis$
$2 Q open parenthesis t close parenthesis$
$Q (t^{2})$
$[ Q open parenthesis t close parenthesis ] squared$

In Problems 65–68, evaluate the function for the given expressions and simplify.

$f (x) = x^{3}$

$f (a^{2})$
$a^{3} \cdot f (a^{3})$
$f (a b)$
$f (a + b)$

$a to the power 6$
$a to the power 12$
$a^{3} b^{3}$
$a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$

$g (x) = x^{4}$

$g (a^{3})$
$a^{4} \cdot g (a^{4})$
$g (a b)$
$g (a + b)$

$F (x) = 3 x^{5}$

$F open parenthesis 2 a close parenthesis$
$2 F open parenthesis a close parenthesis$
$F (a^{2})$
$[ F open parenthesis a close parenthesis ] squared$

$96 a to the power 5$
$6 a to the power 5$
$3 a to the power 10$
$9 a to the power 10$

$G open parenthesis x close parenthesis equals 4 x cubed$

$G open parenthesis 3 a close parenthesis$
$3 G open parenthesis a close parenthesis$
$G open parenthesis a to the power 4 close parenthesis$
$[ G open parenthesis a close parenthesis ] to the power 4$

For the functions in Problems 69–76, compute the following:

$f open parenthesis 2 close parenthesis plus f open parenthesis 3 close parenthesis$
$f open parenthesis 2 plus 3 close parenthesis$
$f (a) + f (b)$
$f (a + b)$

For which functions does $f (a + b) = f (a) + f (b)$ for all values of $a$ and $b$ ?

$f (x) = 3 x - 2$

$11$
$13$
$3 a + 3 b - 4$
$3 a + 3 b - 2$

This function does NOT satisfy $f (a + b) = f (a) + f (b)$ .

$f (x) = 1 - 4 x$

$f (x) = x^{2} + 3$

$19$
$28$
$a^{2} + b^{2} + 6$
$a^{2} + 2 a b + b^{2} + 3$

This function does NOT satisfy $f (a + b) = f (a) + f (b)$ .

$f (x) = x^{2} - 1$

$f (x) = \sqrt{x + 1}$

$\sqrt{3} + 2$
$the square root of 6$
$\sqrt{a + 1} + \sqrt{b + 1}$
$\sqrt{a + b + 1}$

This function does NOT satisfy $f (a + b) = f (a) + f (b)$ .

$f (x) = \sqrt{6 - x}$

$f (x) = \frac{- 2}{x}$

$\frac{- 5}{3}$
$\frac{- 2}{5}$
$\frac{- 2}{a} - \frac{- 2}{b}$
$\frac{- 2}{a + b}$

This function does NOT satisfy $f (a + b) = f (a) + f (b)$ .

$f (x) = \frac{3}{x}$

Use a table of values to estimate a solution to

$f (x) = 800 + 6 x - 0.2 x^{2} = 500$

as follows:

Make a table starting at $x equals 0$ and increasing by $Δ x equals 10$ , as shown in the accompanying tables. Find two $x$ -values $a$ and $b$ so that $f open parenthesis a close parenthesis greater than 500 greater than f open parenthesis b close parenthesis$ .
$x$ $0$ $10$ $20$ $30$ $40$ $50$ $60$ $70$ $80$ $90$ $100$
$f (x)$
Make a new table starting at $x = a$ and increasing by $Δ x equals 1$ . Find two $x$ -values, $c$ and $d$ , so that $f open parenthesis c close parenthesis greater than 500 greater than f open parenthesis d close parenthesis$ .
Make a new table starting at $x = c$ and increasing by $Δ x equals 0.1$ . Find two $x$ -values, $p$ and $q$ , so that $f open parenthesis p close parenthesis greater than 500 greater than f open parenthesis q close parenthesis$ .
Take the average of $p$ and $q$ , that is, set $s = \frac{p + q}{2}$ . Then $s$ is an approximate solution that is off by at most $0.05$ .
Evaluate $f (s)$ to check that the output is approximately $500$ .

$x$ $0$ $10$ $20$ $30$ $40$ $50$ $60$ $70$ $80$ $90$ $100$
$f (x)$ $800$ $840$ $840$ $800$ $720$ $600$ $440$ $240$ $0$ $minus 280$ $minus 600$

$a equals 50$ and $b equals 60$
$x$ $50$ $51$ $52$ $53$ $54$ $55$ $56$ $57$ $58$
$f (x)$ $600$ $585.8$ $571.2$ $556.2$ $540.8$ $525$ $508.8$ $492.2$ $475.2$

$c equals 56$ and $d equals 57$
$x$ $56$ $56.1$ $56.2$ $56.3$ $56.4$ $56.5$ $56.6$
$f (x)$ $508.8$ $507.158$ $505.512$ $503.862$ $502.208$ $500.55$ $498.888$

$p equals 56.5$ and $q equals 56.6$
$s equals 56.55$
$f open parenthesis 56.55 close parenthesis equals 499.7195$

$x$	$0$	$10$	$20$	$30$	$40$	$50$	$60$	$70$	$80$	$90$	$100$
$f (x)$	$800$	$840$	$840$	$800$	$720$	$600$	$440$	$240$	$0$	$minus 280$	$minus 600$

$x$	$50$	$51$	$52$	$53$	$54$	$55$	$56$	$57$	$58$
$f (x)$	$600$	$585.8$	$571.2$	$556.2$	$540.8$	$525$	$508.8$	$492.2$	$475.2$

$x$	$56$	$56.1$	$56.2$	$56.3$	$56.4$	$56.5$	$56.6$
$f (x)$	$508.8$	$507.158$	$505.512$	$503.862$	$502.208$	$500.55$	$498.888$

Use a table of values to estimate a solution to

$f (x) = x^{3} - 4 x^{2} + 5 x = 18, 000$

as follows:

Make a table starting at $x equals 0$ and increasing by $Δ x equals 10$ , as shown in the accompanying tables. Find two $x$ -values $a$ and $b$ so that $f open parenthesis a close parenthesis less than 18, 000 less than f open parenthesis b close parenthesis$ .
$x$ $0$ $10$ $20$ $30$ $40$ $50$ $60$ $70$ $80$ $90$ $100$
$f (x)$
Make a new table starting at $x = a$ and increasing by $Δ x equals 1$ . Find two $x$ -values, $c$ and $d$ , so that $f open parenthesis c close parenthesis less than 18, 000 less than f open parenthesis d close parenthesis$ .
Make a new table starting at $x = c$ and increasing by $Δ x equals 0.1$ . Find two $x$ -values, $p$ and $q$ , so that $f open parenthesis p close parenthesis less than 18, 000 less than f open parenthesis q close parenthesis$ .
Take the average of $p$ and $q$ , that is, set $s = \frac{p + q}{2}$ . Then $s$ is an approximate solution that is off by at most $0.05$ .
Evaluate $f (s)$ to check that the output is approximately $18, 000$ .

Use tables of values to estimate the positive solution to

$f (x) = x^{2} - \frac{1}{x} = 9000$

, accurate to within $0.05$ .

$94.85$

Use tables of values to estimate the positive solution to

$f (x) = \frac{8}{x} + 500 - \frac{x^{2}}{9} = 300$

, accurate to within $0.05$ .

Modeling, Functions, and Graphs by Katherine Yoshiwara (yoshiwarabooks.org), GNU Free Documentation License 1.2 or later. Adapted for the XYZ HTML edition with the authors' permission (recorded 2026-07-04). License: GFDL-1.2-or-later.

Year $(t)$	Total sales $(S)$
2000	$612,000
2001	$663,000
2002	$692,000
2003	$749,000
2004	$904,000