Modeling, Functions, and Graphs

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.

Checkpoint 1.36. QuickCheck 1.

What distinguishes a function from other variable relationships?

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

$\text{C) There ... input value.}$

There cannot be two output values for a single input 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.

Subsection 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 1.37.

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, $S\text{,}$ is a function of $t\text{.}$

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 \lt w \le 1 $	$0.60
$1 \lt w \le 2 $	$0.84
$2 \lt w \le 3 $	$1.08
$3 \lt w \le 4 $	$1.32
$4 \lt w \le 5 $	$1.56
$5 \lt w \le 6 $	$1.80
$6 \lt w \le 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 $w\text{.}$

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	$\alert{51}$	$\alert{209}$
48	232	53	241
55	198	49	186
56	238	$\alert{51}$	$\alert{216}$
$\alert{51}$	$\alert{227}$	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.

Note 1.38.

Note that several different inputs for a function can have the same output. For example, the inputs 4.5 and 4.25 in part (b) of the Example above have output $1.31. However, a single input cannot have more than one output, as illustrated in part (c) of the Example.

Checkpoint 1.39. Practice 2.

Decide whether each table describes $y$ as a function of $x\text{.}$ 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 $x\text{?}$

Each value of $x$ has exactly one value of $y$ associated with it.
For example, $x=3.5$ corresponds both to $y=2.5$ and also to $y=4.0$

$x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$

$y$ $17$ $3$ $0$ $-1$ $0$ $3$ $17$

Is $y$ a function of $x\text{?}$

Each value of $x$ has exactly one value of $y$ associated with it.
For example, $y=3$ corresponds both to $x=-2$ and also to $x=2$

$\text{No}$

$\text{Choice 2}$

$\text{Yes}$

$\text{Choice 1}$

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

Checkpoint 1.40. QuickCheck 2.

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

A) The output values are all the same.
B) The input values are not evely spaced.
C) Two different input values have the same output value.
D) Two different output values have the same input value.

$\text{D) Two ... input value.}$

Two different output values have the same input value.

Subsection 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 1.41.

The graph shows the number of hours, $H\text{,}$ that the sun is above the horizon in Peoria, Illinois, on day $t\text{,}$ where $t = 0$ 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 $H\text{,}$ and when do these values occur?

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

Checkpoint 1.42. Practice 3.

The graph shows the elevation in feet, $a\text{,}$ 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?
- The input variable is $d \text{,}$ and the output variable is $a \text{.}$
- The input variable is $a \text{,}$ and the output variable is $d \text{.}$
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 $a \text{,}$ 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

$\text{Choice 1}$

$210$

Answer 3.

$5, 11, 12, 16, 17.5, 18$

$300$

$0$

$165$

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

Subsection Functions Defined by Equations

Example 1.43 illustrates a function defined by an equation.

Example 1.43.

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

\begin{equation*} h = 1776 - 16t^2 \end{equation*}

Thus, after $\alert{1}$ second the book’s height is

\begin{equation*} h = 1776 - 16(\alert{1})^2 = 1760 \text{ feet} \end{equation*}

After $\alert{2}$ seconds its height is

\begin{equation*} h = 1776 - 16(\alert{2})^2 = 1712 \text{ feet} \end{equation*}

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

Checkpoint 1.44. Practice 4.

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

$V=$

${\frac{4}{3}}\pi r^{3}$

$V=\dfrac{4}{3}\pi r^3$

Checkpoint 1.45. QuickCheck 3.

Name three ways to describe a function.

A) By inputs, outputs, or evaluation
B) By tables, equations, or graphs
C) By the intercepts, the slope, or the vertex
D) By numbers, letters, or diagrams

$\text{B) By ... , or graphs}$

By tables, equations, or graphs

Technology 1.46. 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 1.43,

\begin{equation*} h = 1776 - 16t^2 \end{equation*}

we follow the steps:

Enter the equation: Press the Y= key, clear out any other equations, and define $Y_1 = 1776 - 16X^2.$
Choose the $x$-values for the table. Press 2ndWINDOW to access the $TblSet$ (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 $(TblStart = 0)$ and an increment of one unit in the $x$-values, $(\Delta Tbl = 1)\text{.}$ 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 1.43.

Now try making a table of values with $TblStart = 0$ and $\Delta Tbl = 0.5\text{.}$ Use the ↑ and ↓ arrow keys to scroll up and down the table.

Checkpoint 1.47. Pause and Reflect.

Write one question you still have about functions.

Subsection Function Notation

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

For instance, in Example 1.43, the height, $h\text{,}$ of a falling algebra book is a function of the elapsed time, $t\text{.}$ We might call this function $f\text{.}$ In other words, $f$ is the name of the relationship between the variables $h$ and $t\text{.}$ We write

\begin{equation*} h = f (t) \end{equation*}

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

Caution 1.48.

The new symbol $f(t)\text{,}$ read "$f$ of $t\text{,}$" is another name for the height, $h\text{.}$ The parentheses in the symbol $f(t)$ do not indicate multiplication. (It would not make sense to multiply the name of a function by a variable.) Think of the symbol $f(t)$ as a single variable that represents the output value of the function.

With this new notation we may write

\begin{equation*} h = f (t) = 1776 - 16t^2 \end{equation*}

or just

\begin{equation*} f (t) = 1776 - 16t^2 \end{equation*}

instead of

\begin{equation*} h = 1776 - 16t^2 \end{equation*}

to describe the function.

Note 1.49.

Perhaps it seems complicated to introduce a new symbol for $h\text{,}$ but the notation $f(t)$ is very useful for showing the correspondence between specific values of the variables $h$ and $t\text{.}$

Example 1.50.

In Example 1.43, the height of an algebra book dropped from the top of One World Trade Center is given by the equation

\begin{equation*} h = 1776 - 16t^2 \end{equation*}

We see that

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

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

\(f(1)=1760\)
and
\(f(2)=1712\)

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

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

Function Notation.

Checkpoint 1.51. QuickCheck 4.

True or False.

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

$\text{False}$

$\text{False}$

$\text{False}$

$\text{False}$

False
False
False
False

Checkpoint 1.52. Practice 5.

Let $F $ be the name of the function defined by the graph in Example 1.41, the number of hours of daylight in Peoria $t$ days after January 1.

Use function notation to state that $H$ is a function of $t\text{.}$
- $\displaystyle F=H(t) $
- $\displaystyle H=F(t) $
- $\displaystyle t=F(H) $
- $\displaystyle H=t(F) $
What does the statement $F(15) = 9.7$ mean in the context of the problem?
- A) The sun is 9.7 degrees above the horizon in Peoria on January 15.
- B) The sun is above the horizon in Peoria for 15 hours on January 10.
- C) The sun is above the horizon in Peoria for 9.7 hours on January 16.

$\text{Choice 2}$

$\text{C) The ... January 16.}$

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

Checkpoint 1.53. QuickCheck 4.

Use function notation to write the statement “$L$ defines $w$ as a function of $p\text{.}$”

$\displaystyle L=w(p)$
$\displaystyle w=L(p)$
$\displaystyle p=L(w)$
$\displaystyle L=p(w)$

$\text{Choice 2}$

$w=L(p)$

Subsection 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 1.54.

Let $g$ be the name of the postage function defined by the table in Example 1.34 b. Find $g(1)\text{,}$ $g(3)\text{,}$ and $g(6.75$).

According to the table,

when $w=1\text{,}$	$p=0.47$	so	$g(1)=0.47$
when $w=3\text{,}$	$p=0.89$	so	$g(3)=0.89$
when $w=6.75\text{,}$	$p=1.73$	so	$g(6.75)=1.73$

Thus, a letter weighing 1 ounce costs $0.47 to mail, a letter weighing 3 ounces costs $0.89, and a letter weighing 6.75 ounces costs $1.73.

Checkpoint 1.55. Practice 6.

When you exercise, your heart rate should increase until it reaches your target heart rate. The table shows target heart rate, $r = f(a) \text{,}$ 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 $f(25)$ and $f(50) \text{.}$

$f(25)=$

$f(50)=$
Find a value of $a$ for which $f(a) = 135 \text{.}$

$a=$

$146$

$127$

$40$

$f (25) = 146 \text{,}$ $f(50) = 127$
$\displaystyle a = 40$

Checkpoint 1.56. QuickCheck 5.

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

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

$\text{Choice 3}$

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

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 1.57.

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

$\displaystyle s=6$
$\displaystyle s=-1$

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

Checkpoint 1.58. Practice 7.

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

$x$	$f(x)$
$-2$		$f(\alert{-2})=5-(\alert{-2})^3=~$
$0$		$f(\alert{0})=5-\alert{0}^3=$
$1$		$f(\alert{1})=5-\alert{1}^3=$
$3$		$f(\alert{3})=5-\alert{3}^3=$

$13$

$5$

$4$

$-22$

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

Technology 1.59. Evaluating a Function.

We can use the table feature on a graphing calculator to evaluate functions. Consider the function of Checkpoint 1.58, $f(x) = 5 - x^3\text{.}$

Press Y=, clear any old functions, and enter

$\qquad Y_1=5-X$ ^ $3$
Press TblSet (2nd WINDOW) and choose $Ask$ after $Indpnt\text{,}$ 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 Checkpoint 1.58, 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 $Y_1$ at the top of the second column. The definition of $Y_1$ 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.

Example 1.60.

TrailGear decides to market a line of backpacks. The cost, $C\text{,}$ of manufacturing backpacks is a function of the number, $x\text{,}$ of backpacks produced, given by the equation

\begin{equation*} C(x) = 3000 + 20x \end{equation*}

where $C(x)$ is measured in dollars. Find the cost of producing 500 backpacks.

To find the value of $C$ that corresponds to $x = \alert{500}\text{,}$ evaluate $C(500)\text{.}$

\begin{equation*} C(\alert{500}) = 3000 + 20(\alert{500}) = 13,000 \end{equation*}

The cost of producing 500 backpacks is $13,000.

Checkpoint 1.61. Practice 8.

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

\begin{equation*} V = V(r) = \frac{4}{3}\pi r^3 \end{equation*}

Evaluate $V(10)$ and explain what it means.

$V(10)=$, which represents

A) the volume (in cm) of a sphere whose radius is 10 cu. cm
B) the radius (in cm) of a sphere whose volume is 10 cm
C) the volume (in sq. cm) of a sphere whose radius is 10 cm
D) the volume (in cu. cm) of a sphere whose radius is 10 cm

$4188.79$

$\text{D) the ... is 10 cm}$

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

Checkpoint 1.62. Pause and Reflect.

Do linear equations $y=mx+b$ and quadratic equations $y=ax^2+bx+c$ define functions? Why or why not?

Subsection Operations with Function Notation

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

Example 1.63.

TrailGear manufactures backpacks at a cost of

\begin{equation*} C(x) = 3000 + 20x \end{equation*}

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, $b\text{.}$ Therefore, a co-op that produced $b$ backpacks per month during the winter should increase production to $b + 0.5b\text{,}$ or $1.5b$ backpacks per month in the summer.
The cost of producing $1.5b$ backpacks will be
\begin{equation*} C(\alert{1.5b}) = 3000 + 20(\alert{1.5b}) = 3000 + 30b \end{equation*}

Checkpoint 1.64. Practice 9.

A spherical balloon has a radius of 10 centimeters.

If we increase the radius by $h$ centimeters, what will the new volume be?
- $V(10)+h=\frac{4}{3}\pi(10^3)+h $ cu. cm
- $V(10)+h=\frac{4}{3}\pi(10^{3+h}) $ cu. cm
- $V(10+h)=\frac{4}{3}\pi\cdot 10^3+h $ cu. cm
- $V(10+h)=\frac{4}{3}\pi(10+h)^3 $ cu. cm
If $h = 2 \text{,}$ how much did the volume increase? Round your answer to hundredths.

It increased by $\text{cm}^3 \text{.}$

$\text{Choice 4}$

$3049.44$

$\displaystyle V(10 + h) = \dfrac{4}{3}\pi(10 + h)^3 \text{ cm}^3$
From $V(10)$ to $V(12)$ is an increase of about $3049.44 \text{ cm}^3$

Example 1.65.

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

$\displaystyle x = 2h$
$\displaystyle x = a + 3$

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

Caution 1.66.

In Example 1.65, notice that

\begin{equation*} f(2h) \ne 2 f(h) \end{equation*}

and

\begin{equation*} f(a + 3) \ne f(a) + f(3) \end{equation*}

To compute $f(a) + f(3)\text{,}$ we must first compute $f(a)$ and $f(3)\text{,}$ then add them:

\begin{equation*} \begin{aligned}[t] f(a)+f(3)\amp= (4a^2-a+5)+(4\cdot 3^2-3+5) \\ \amp= 4a^2 - a + 43\\ \end{aligned} \end{equation*}

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

Checkpoint 1.67. QuickCheck 6.

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

$\displaystyle f(x)+f(y)$
$\displaystyle f(x+y)^2$
$\displaystyle x^2+y^2$
$\displaystyle (x+y)^2$

$\text{Choice 4}$

$(x+y)^2$

Checkpoint 1.68. Practice 10.

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

$f(2) + f(3)=$
$f(2 + 3)=$
$2 f(x) + 3$

$33$

$124$

Answer 3.

$2\!\left(x^{3}-1\right)+3$