## Section1.2Functions

### SubsectionDefinition 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

\begin{equation*} C=800+30t ~~~~ (t\ge 0) \end{equation*}

Thus, for example

 when $t=\alert{0}\text{,}$ $C=800+30(\alert{0})=800$ when $t=\alert{4}\text{,}$ $C=800+30(\alert{4})=920$ when $t=\alert{10}\text{,}$ $C=800+30(\alert{10})=1100$

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 $t\text{.}$ 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.

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

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

###### Definition of Function

A function is a relationship between two variables for which a unique value of the output variable can be determined from a value of the input variable.

###### Note1.21

What distinguishes functions from other variable relationships? The definition of a function calls for a unique value—that is, exactly one value of the output variable corresponding to each value of the input variable. This property makes functions useful in applications because they can often be used to make predictions.

###### Example1.22
1. The distance, $d\text{,}$ traveled by a car in 2 hours is a function of its speed, $r\text{.}$ If we know the speed of the car, we can determine the distance it travels by the formula $d = r \cdot 2\text{.}$

2. The cost of a fill-up with unleaded gasoline is a function of the number of gallons purchased. The gas pump represents the function by displaying the corresponding values of the input variable (number of gallons) and the output variable (cost).

3. Score on the Scholastic Aptitude Test (SAT) is not a function of score on an IQ test, because two people with the same score on an IQ test may score differently on the SAT; that is, a person’s score on the SAT is not uniquely determined by his or her score on an IQ test.

###### Checkpoint1.23
1. 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.

2. 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.

1. No, students with the same grade in English can have different grades in algebra.

2. Yes, the number of calories is proportional to the number of fluid ounces.

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

### SubsectionFunctions 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.

###### Example1.24
1. 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 2. The table below gives the cost of sending printed material by first-class mail in 2016.  Weight in ounces $(w)$ Postage $(P)$ $0 \lt w \le 1$$0.47 $1 \lt w \le 2$ $0.68 $2 \lt w \le 3$$0.89 $3 \lt w \le 4$ $1.10 $4 \lt w \le 5$$1.31 $5 \lt w \le 6$ $1.52 $6 \lt w \le 7$$1.73

###### Checkpoint1.36

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.

 $a$ $20$ $25$ $30$ $35$ $40$ $45$ $50$ $55$ $60$ $65$ $70$ $r$ $150$ $146$ $142$ $139$ $135$ $131$ $127$ $124$ $120$ $116$ $112$
1. Find $f(25)$ and $f(50)\text{.}$

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

1. $f (25) = 146, ~~f (50) = 127$

2. $a = 40$

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.

###### Example1.37

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

1. $s=6$

2. $s=-1$

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

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

###### Checkpoint1.38

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 $x\text{.}$

 $x$ $f(x)$ $-2$  $f(\alert{-2})=5-(\alert{-2})^3=~$ $0$  $f(\alert{0})=5-\alert{0}^3=$ $1$  $f(\alert{1})=5-\alert{1}^3=$ $3$  $f(\alert{3})=5-\alert{3}^3=$
 $x$ $f(x)$ $-2$ $13$ $0$ $5$ $1$ $4$ $3$ $-22$
###### Technology1.39Evaluating a Function

We can use the table feature on a graphing calculator to evaluate functions. Consider the function of Checkpoint 1.38, $f(x) = 5 - x^3\text{.}$

• 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 1.38, 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 $\boxed{\rightarrow}$ and $\boxed{\uparrow}$ 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.

###### Example1.40

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.

Solution

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

###### 26
 Cost ofmerchandise ($M$) Shippingcharge ($C$) $\0.01-10.00$ $\2.50$ $10.01-20.00$ $3.75$ $20.01-35.00$ $4.85$ $35.01-50.00$ $5.95$ $50.01-75.00$ $6.95$ $75.01-100.00$ $7.95$ Over $100.00$ $8.95$
###### 27

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

1. $g(25)$

2. $g(40)$

3. $x$ so that $g(x) = 50$

###### 28

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

1. $h(20)$

2. $h(60)$

3. $x$ so that $h(x) = 10$

###### 29

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

1. $T(8750)$

2. $T(6249)$

3. $x$ so that $T(x) = 15\%$

###### 30

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

1. $C(11.50)$

2. $C(47.24)$

3. $x$ so that $C(x) = 7.95$

###### 31

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 ofwomen $47.5$ $51.4$ $47.0$ $62.5$ $66.2$ $67.7$ $68.3$ $65.5$ $66.0$
1. Evaluate $f (1985)$ and explain what it means.

2. Estimate a solution to the equation $f (t) = 68$ and explain what it means.

3. 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.

###### 32

The table shows $f (t)\text{,}$ the death rate (per 100,000 people) from HIV among 15–24-year-olds, and $g(t)\text{,}$ 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 $11.7$ $14.0$ $17.9$ $19.7$ $24.2$ $28.6$ $19.2$ $8.1$ $6.1$ $4.6$
1. Evaluate $f (1995)$ and explain what it means.

2. Find a solution to the equation $g (t) = 28.6$ and explain what it means.

3. 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.

###### 33

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.

 $a$ $20$ $25$ $30$ $35$ $40$ $45$ $50$ $55$ $60$ $65$ $70$ $r$ $150$ $146$ $142$ $139$ $135$ $131$ $127$ $124$ $120$ $116$ $112$
1. Does $f (50) = 2 f (25)\text{?}$

2. Find a value of a for which $f (a) = 2a\text{.}$ Is $f (a) = 2a$ for all values of $a\text{?}$

3. Is $r = f (a)$ an increasing function or a decreasing function?

###### 34

The table shows $M = f (d)\text{,}$ the men's Olympic record time, and $W = g(d)\text{,}$ the women's Olympic record time, as a function of the length, $d\text{,}$ of the race. For example, the women’s record in the 100 meters is 10.62 seconds, and the men’s record in the 800 meters is 1 minute, 42.58 seconds. (Source: www.hickoksports.com)

 Distance(meters) $100$ $200$ $400$ $800$ $1500$ $5000$ $10,000$ Men $9.63$ $19.30$ $43.03$ $1:40.91$ $3:32.07$ $12:57.82$ $27:01.17$ Women $10.62$ $21.34$ $48.25$ $1:53.43$ $3:53.96$ $14:26.17$ $29:17.45$
1. Does $f (800) = 2 f (400)\text{?}$ Does $g(400) = 2g(200)\text{?}$

2. Find a value of $d$ for which $f (2d)\lt 2f (d)\text{.}$ Is there a value of $d$ for which $g(2d)\lt 2g(d)\text{?}$

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

###### 35

The graph shows $C$ as a function of $t\text{.}$ $C$ 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. 1. When did $2000$ students consider themselves computer literate?

2. How long did it take that number to double?

3. How long did it take for the number to double again?

4. How many students became computer literate between January 1992 and June 1993?

###### 36

The graph shows $P$ as a function of $t\text{.}$ $P$ is the number of people in Cedar Grove who owned a portable DVD player $t$ years after 2000. 1. When did 3500 people own portable DVD players?

2. How many people owned portable DVD players in 2005?

3. The number of owners of portable DVD players in Cedar Grove seems to be leveling off at what number?

4. How many people acquired portable DVD players between 2001 and 2004?

###### 37

The graph shows the revenue, $R\text{,}$ a movie theater collects as a function of the price, $d\text{,}$ it charges for a ticket. 1. What is the revenue if the theater charges $$12.00$ for a ticket? 2. What should the theater charge for a ticket in order to collect$$1500$ in revenue?

3. For what values of $d$ is $R\gt 1875\text{?}$

###### 38

The graph shows $S$ as a function of $w\text{.}$ $S$ represents the weekly sales of a best-selling book, in thousands of dollars, $w$ weeks after it is released. 1. In which weeks were sales over $$7000\text{?}$ 2. In which week did sales fall below$$5000$ on their way down?

3. For what values of $w$ is $S\gt 3.4\text{?}$

###### 39

The graph shows the federal minimum wage, $M\text{,}$ as a function of time, $t\text{,}$ adjusted for inflation to reflect its buying power in 2004 dollars. (Source: www.infoplease.com) 1. When did the minimum wage reach its highest buying power, and what was it worth in 2004 dollars?

2. When did the minimum wage fall to its lowest buying power after its peak, and what was its worth at that time?

3. Give two years in which the minimum wage was worth $$8$ in 2004 dollars. ###### 40 The graph shows the U.S. unemployment rate, $U\text{,}$ as a function of time, $t\text{,}$ for the years 1985–2004. (Source: U.S. Bureau of Labor Statistics) 1. When did the unemployment rate reach its highest value, and what was its highest value? 2. When did the unemployment rate fall to its lowest value, and what was its lowest value? 3. Give two years in which the unemployment rate was $4.5\%\text{.}$ In Problems 41–48, evaluate each function for the given values. ###### 41 $f (x) = 6 - 2x$ 1. $f(3)$ 2. $f(-2)$ 3. $f(12.7)$ 4. $f\left(\dfrac{2}{3}\right)$ ###### 42 $g(t) = 5t - 3$ 1. $g(1)$ 2. $g(-4)$ 3. $g(14.1)$ 4. $g\left(\dfrac{3}{4}\right)$ ###### 43 $h(v) = 2v^2 - 3v + 1$ 1. $h(0)$ 2. $h(-1)$ 3. $h\left(\dfrac{1}{4}\right)$ 4. $h(-6.2)$ ###### 44 $r (s) = 2s - s^2$ 1. $r(2)$ 2. $r(-4)$ 3. $r\left(\dfrac{1}{3}\right)$ 4. $r(-1.3)$ ###### 45 $H(z) = \dfrac{2z - 3}{z + 2}$ 1. $H(4)$ 2. $H(-3)$ 3. $H\left(\dfrac{4}{3}\right)$ 4. $H(4.5)$ ###### 46 $F(x) = \dfrac{1-x}{2x-3}$ 1. $F(0)$ 2. $F(-3)$ 3. $F\left(\dfrac{5}{2}\right)$ 4. $F(9.8)$ ###### 47 $E(t) =\sqrt{t-4}$ 1. $E(16)$ 2. $E(4)$ 3. $E(7)$ 4. $E(4.2)$ ###### 48 $D(r) =\sqrt{5-r}$ 1. $D(4)$ 2. $D(-3)$ 3. $D(-9)$ 4. $D(4.6)$ ###### 49 A sport utility vehicle costs$$28,000$ and depreciates according to the formula

\begin{equation*} V(t) = 28,000 (1 - 0.08t) \end{equation*}

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

1. Evaluate $V(12)$ and explain what it means.

2. Solve the equation $V(t) = 0$ and explain what it means.

3. If this year is $t = n\text{,}$ what does $V(n + 2)$ mean?

###### 50

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

\begin{equation*} S(x) = 20,000 + 0.01x \end{equation*}

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

1. Evaluate $S(850,000)$ and explain what it means.

2. Solve the equation $S(x) = 30,000$ and explain what it means.

3. If the company made a profit of $p$ dollars this year, what does $S(2p)$ mean?

###### 51

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

\begin{equation*} N( p) = \dfrac{12,000,000}{p} \end{equation*}
1. Evaluate $N(6000)$ and explain what it means.

2. As $p$ increases, does $N(p)$ increase or decrease? Why is this reasonable?

3. If the current price for a compact car is $D\text{,}$ what does $2N(D)$ mean?

###### 52

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

\begin{equation*} M(t) = \dfrac{600,000}{t},~~ t\le 30 \end{equation*}

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

1. Evaluate $M(2)$ and explain what it means.

2. As $t$ increases, does $M(t)$ increase or decrease? Why is this reasonable?

3. If this week $t = n\text{,}$ what does $M(n + 1)$ mean?

###### 53

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

\begin{equation*} v(d) = \sqrt{12d} \end{equation*}

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

1. Evaluate $v(250)$ and explain what it means.

2. Estimate the length of the skid marks left by a car traveling at $100$ miles per hour.

###### 54

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

\begin{equation*} d(h) = 1.22\sqrt{h} \end{equation*}
1. Evaluate $d(20,320)$ and explain what it means.

2. Estimate the height you need in order to see $100$ miles.

###### 55

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) 1. During June and July, avalanches occurred over three separate time intervals. What were they?

2. Over what three time intervals did snow fall?

3. When was the temperature above freezing ($0\degree$C)?

###### 56

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) 1. Read the graph and complete the table.

 Altitude (km) Percent ofEarth's surface $-7$ to $-6$  $-6$ to $-5$  $-5$ to $-4$  $-4$ to $-3$  $-3$ to $-2$  $-2$ to $-1$  $-1$ to $0$  $0$ to $1$  $1$ to $2$  $2$ to $3$  $3$ to $4$  $4$ to $5$ 
2. What is the most common altitude? What is the second most common altitude??

3. Approximately what percent of the Earth's surface is below sea level?

4. The height of Mt. Everest is $8.85$ kilometers. Can you think of a reason why it is not included in the graph?

###### 57

The graph shows the temperature of the ocean at various depths. (Source: Open University) 1. Is depth a function of temperature?

2. Is temperature a function of depth?

3. The axes are scaled in an unusual way. Why is it useful to present the graph in this way?

###### 58

The graph shows the relationship between annual precipitation, $p\text{,}$ in a region and the amount of erosion, measured in tons per square mile, $s\text{.}$ (Source: Leopold, Wolman, Miller, 1992) 1. Is the amount of erosion a function of the amount of precipitation?

2. At what annual precipitation is erosion at a maximum, and what is that maximum?

3. Over what interval of annual precipitation does erosion decrease?

4. 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: $0\lt p\lt 12$ grassland: $12\lt p\lt 30$ forest: $30\lt p\lt 60$

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

###### 59

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

1. $G(3a)$

2. $G(a + 2)$

3. $G(a) + 2$

4. $G(-a)$

###### 60

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

1. $h(2a)$

2. $h(a + 3)$

3. $h(a) + 3$

4. $h(-a)$

###### 61

$g(x) = 8$

1. $g(2)$

2. $g(8)$

3. $g(a + 1)$

4. $g(-x)$

###### 62

$f (t) = -3$

1. $f (4)$

2. $f (-3)$

3. $f (b - 2)$

4. $f (-t)$

###### 63

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

1. $P(2x)$

2. $2P(x)$

3. $P(x^2)$

4. $[P(x)]^2$

###### 64

$Q(t) = 5t^3$

1. $Q(2t)$

2. $2Q(t)$

3. $Q(t^2)$

4. $[Q(t)]^2$

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

###### 65

$f (x) = x^3$

1. $f (a^2)$

2. $a^3 \cdot f (a^3)$

3. $f (ab)$

4. $f (a + b)$

###### 66

$g(x) = x^4$

1. $g(a^3)$

2. $a^4\cdot g(a^4)$

3. $g(ab)$

4. $g(a + b)$

###### 67

$F(x) = 3x^5$

1. $F(2a)$

2. $2 F(a)$

3. $F(a^2)$

4. $[F(a)]^2$

###### 68

$G(x) = 4x^3$

1. $G(3a)$

2. $3G(a)$

3. $G(a^4)$

4. $[G(a)]^4$

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

1. $f (2) + f (3)$

2. $f (2 + 3)$

3. $f (a) + f (b)$

4. $f (a + b)$

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

###### 69

$f (x) = 3x - 2$

###### 70

$f (x) = 1 - 4x$

###### 71

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

###### 72

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

###### 73

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

###### 74

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

###### 75

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

###### 76

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

###### 77

Use a table of values to estimate a solution to

\begin{equation*} f (x) = 800 + 6x - 0.2x^2 = 500 \end{equation*}

as follows:

1. Make a table starting at $x = 0$ and increasing by $\Delta x = 10\text{,}$ as shown in the accompanying tables. Find two $x$-values $a$ and $b$ so that $f (a)\gt 500\gt f (b)\text{.}$

 $x$ $0$ $10$ $20$ $30$ $40$ $50$ $60$ $70$ $80$ $90$ $100$ $f(x)$           
2. Make a new table starting at $x = a$ and increasing by $\Delta x = 1\text{.}$ Find two $x$-values, $c$ and $d\text{,}$ so that $f (c)\gt 500\gt f (d)\text{.}$

3. Make a new table starting at $x = c$ and increasing by $\Delta x = 0.1\text{.}$ Find two $x$-values, $p$ and $q\text{,}$ so that $f (p)\gt 500\gt f (q)\text{.}$

4. Take the average of $p$ and $q\text{,}$ that is, set $s = \dfrac{p + q}{2}\text{.}$ Then $s$ is an approximate solution that is off by at most $0.05\text{.}$

5. Evaluate $f (s)$ to check that the output is approximately $500\text{.}$

###### 78

Use a table of values to estimate a solution to

\begin{equation*} f (x) = x^3 - 4x^2 + 5x = 18, 000 \end{equation*}

as follows:

1. Make a table starting at $x = 0$ and increasing by $\Delta x = 10\text{,}$ as shown in the accompanying tables. Find two $x$-values $a$ and $b$ so that $f (a)\lt 18,000\lt f (b)\text{.}$

 $x$ $0$ $10$ $20$ $30$ $40$ $50$ $60$ $70$ $80$ $90$ $100$ $f(x)$           
2. Make a new table starting at $x = a$ and increasing by $\Delta x = 1\text{.}$ Find two $x$-values, $c$ and $d\text{,}$ so that $f (c)\lt 18,000\lt f (d)\text{.}$

3. Make a new table starting at $x = c$ and increasing by $\Delta x = 0.1\text{.}$ Find two $x$-values, $p$ and $q\text{,}$ so that $f (p)\lt 18,000\lt f (q)\text{.}$

4. Take the average of $p$ and $q\text{,}$ that is, set $s = \dfrac{p + q}{2}\text{.}$ Then $s$ is an approximate solution that is off by at most $0.05\text{.}$

5. Evaluate $f (s)$ to check that the output is approximately $18,000\text{.}$

###### 79

Use tables of values to estimate the positive solution to

\begin{equation*} f (x) = x^2 - \dfrac{1}{x} = 9000\text{,} \end{equation*}

accurate to within $0.05\text{.}$

###### 80

Use tables of values to estimate the positive solution to

\begin{equation*} f (x) = \dfrac{8}{x}+500-\dfrac{x^2}{9} = 300\text{,} \end{equation*}

accurate to within $0.05\text{.}$