Stewart J. College Algebra: Concepts and Contexts

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114 CHAPTER 1

■

Data, Functions, and Models

The Vertical Line Test states the fact that a relation is a function if and only if

its graph has the property that no vertical line intersects the graph more than once.

We can represent a function in four different ways:

■

Verbally, using words

■

Numerically, using a table of two-variable data

■

Symbolically, using an equation

■

Graphically, using a graph in the coordinate plane

1.5 Function Notation: The Concept of Function as a Rule

A function can be thought of as a rule that produces exactly one output for every in-

put. To help us describe the rule of a function, we use function notation. If a func-

tion with the name “f ” acts on the input x to produce the output y, then we write

For example, if then f squares each input, so that, for instance,

and .

The net change in the value of a function f from the input a to the input b

(where ) is the difference

The domain of a function is the set of all inputs. If the domain is not explicitly

given, then we assume that the domain is the set of all real numbers for which the

output is defined.

A piecewise defined function is one that is defined by different rules on differ-

ent parts of its domain. For example, the absolute value function is a piecewise de-

fined function:

1.6 Working with Functions: Graphs and Graphing Calculators

The graph of the function f is the set of all ordered pairs (x, y), where , plot-

ted in a coordinate plane. You should be familiar with the graphs of basic functions

such as , , , , , and .

In using a graphing calculator to graph a function, it’s important to select an

appropriate viewing rectangle (“window”) to display the essential features of the

graph properly.

To graph a piecewise defined function, graph each “piece” separately over the

appropriate portion of the domain.

1.7 Working with Functions: Getting Information from the Graph

We can read the behavior or “life history” of a function from its graph, keeping in

mind that the height of the graph at each point represents the value of the function at

that point. From the graph we can determine:

f 1x 2= 1x

f 1x 2= 0 x 0f 1x 2= x

f 1x 2= x

f 1x 2= xf 1x 2= c

y = f 1x2

0 x 0= e

x if x Ú 0

- x if x 6 0

f 1b 2- f 1a2

a … b

f 1- 22= 4

f 132= 9f 1x 2= x

f 1x2= y

■

Values of the function at specific inputs

■

Net change in the value of the function between two inputs

■

The domain and range of the function

■

Intervals on which the function is increasing or decreasing

■

Local maximum and minimum values of the function

■

Where two functions have the same value

■

The intervals where the values of one function are greater than (or less than)

another

1.8 Working with Functions:

Modeling Real-World Relationships

Functions can be used to model real-world relationships. In this section we construct

models from verbal descriptions of the relationships (whereas in Section 1.3 we con-

structed models from data).

Once we have found a function that models a real-world situation, we can use

the model to answer questions and make predictions about the situation being mod-

eled. For instance, we can determine the time when the level of water in a reservoir

will drop to a given depth or the maximum height that will be achieved by a missile

fired from a cannon.

1.9 Making and Using Formulas

A formula is an equation that involves two or more variables. Some familiar formu-

las are , , , and . Formulas are generally used

to describe relationships between real-world quantities. When using formulas, we

often need to solve for one of the variables in terms of the others.

Some formulas use subscripted variables to denote related quantities. For in-

stance, the average A of the three numbers , , and is given by the formula

A =

+ n

P = 2l + 2wC = 2prA = pr

E = mc

CHAPTER 1

■

Review Exercises 115

CHAPTER

SKILLS

REVIEW EXERCISES

1–2 ■ A list of one-variable data is given.

(a) Find the average of the data.

(b) Find the median of the data.

3–4

■ Two-variable data are given.

(a) Express the data as a set of ordered pairs, where x is the input and y is the output of

the relation.

(b) Find the domain and range of the relation.

x 10.3 6.5 7.2 4.6 9.1 8.4 5.0

x 12 4 10

- 4

4 0

11–12 ■ A set of data is given.

(a) Find the first differences to see whether a linear model is appropriate.

(b) If a linear model is appropriate, then find the model.

(d) Make a scatter plot of the data.

(e) Does there appear to be a relationship between the variables? If so, describe the

relationship.

3. 4.

5–6

■ The graph of a relation is given.

(a) List four of the ordered pairs in the relation.

(b) Find the output(s) corresponding to the input 8.

(d) What are the maximum and minimum outputs?

5. 6.

x 2 4 4 6 10 11 12

y 9 1 4 7 3 15 6

x 2 4 6 8 10 12

y 14 12 12 8 5 3

116 CHAPTER 1

■

Data, Functions, and Models

281046

7–10 ■ A table or a scatter plot of a data set is given.

(a) Find a linear model for the data.

(b) Use the model to predict the output for the inputs 3.5 and 30.

7. 8.

9. 10.

x 0 1 2 3 4 5

y 8 5 2

- 1 - 4 - 7

x 0 2 4 6 8 10

y 6 10 14 18 22 26

2315746

8124202416

CHAPTER 1

■

Review Exercises 117

11. 12.

13–14

■ A set of ordered pairs defining a relation is given. Is the relation a function?

13.

14.

15–16

■ Two-variable data are given in the table.

(a) Is the variable y a function of the variable x? If so, which is the independent

variable and which is the dependent variable?

(b) Is the variable x a function of the variable y? If so, which is the independent

variable and which is the dependent variable?

15. 16.

17–18

■ An equation is given in function form.

(a) What is the independent variable and what is the dependent variable?

(b) Find the value of the dependent variable when the independent variable is 4.

changes from 1 to 4.

17. 18.

19–20

■ An equation and its graph are given.

(a) Find the x- and y-intercepts.

(b) Use the Vertical Line Test to see whether the equation defines y as a function of x.

y when x changes from to 1.

19. 20. x

+ 9y

= 9x

- y - 4x = 0

- 3

z = 51t

- 2ty = 3x + x

x 1

- 1

- 2

- 3

- 4

y 1 1 4 4 9 9 16 16

x 1 2 3 2 1

- 2 - 1

0 1 2

511, 32, 11, - 32, 12, - 12, 12, 12, 13, 22, 14, - 32, 15, 526

511, 32, 12, 32, 13, - 12, 15, 2 2, 17, 22, 19, - 126

x y

First

differences

0 56 —

2 60

4 65

6 71

x y

First

differences

0 10.7 —

1 8.4

2 6.1

3 3.8

21–24 ■ An equation in two variables is given.

(a) Does the equation define y as a function of x?

(b) Does the equation define x as a function of y?

21. 22.

23. 24.

25–28

■ A verbal rule describing a function is given.

(a) Give an algebraic representation of the function (an equation).

(b) Give a numerical representation of the function (a table of values).

(d) Express the rule in function notation.

25. “Multiply by 2, then subtract 7.” 26. “Add 3, then divide by 4.”

27. “Add 2, square, then divide by 3.” 28. “Square, subtract 1, then multiply by .”

29–32

■ A function is given.

(a) Give a verbal description of the function.

(b) Express the function as an equation. Identify the dependent and the independent

variable.

29. 30.

31. 32.

33–36

■ Find the indicated values of the given function.

33.

(a) (b) (c) (d)

34.

(a) (b) (c) (d)

35.

(a) (b) (c) (d)

36.

(a) (b) (c) (d)

37–38

■ A function is given. Complete the table of values, and then graph the function.

37. 38. g1x 2= x

- 4f 1x 2= 5 - 2x

k11.52k112k102k1- 22

k1x 2= e

-x if x … 1

if x 7 1

h11002h112h10 2h1- 42

h1x 2= e

2if x 6 0

3x if x Ú 0

g1b 2g152g1- 32g102

g1x 2= 2x

+ 4x + 4

f 1a 2f 1- 12f 12 2f 102

f 1x 2= 2x

- 4x + 3

k1x 2= 2x

h1x 2=

1x - 32

g1x 2= 71x - 5 2f 1x2=

+ 5

- 6y

- 6 = 021x + 3y = 0

+ y

= 165x - y

= 10

118 CHAPTER 1

■

Data, Functions, and Models

- 2 - 1

01234

f(x)

- 3 - 2 - 1

0 1 2 3

g(x)

39–44 ■ Sketch a graph of the function by first making a table of values.

39. 40.

41. 42.

43. 44. k1x 2= 2 - 1x

k1x 2= 1x + 2

h1x 2= x

- 4x,- 1 … x … 5h1x 2= 4 - x

,- 1 … x … 2

g1x 2=-x + 2g1x 2= 3 +

CHAPTER 1

■

Review Exercises 119

45–48 ■ Draw a graph of the function in an appropriate viewing rectangle.

45. 46.

47. 48.

49–50 ■ Sketch a graph of the piecewise-defined function.

49. 50.

51–52

■ The graph of a function f is given.

(a) Find , , and .

(b) Find the net change in the value of f when x changes from to 1.

(d) Find the domain and range of f.

(e) Find the intervals on which f is increasing and on which it is decreasing.

51. 52.

f 1x 2= 1

- 1

f 122f 102f 1- 12

g1x 2= e

- x if x 6 1

2x if x Ú 1

f 1x 2= e

3 if x 6-1

x if x Ú-1

g1x 2= 4x - x

g1x 2= 2 - 1x + 1

f 1x 2= x

- 2x - 2f 1x 2= 5 + 4x - x

53–54 ■ A function f is given.

(a) Use a graphing calculator to draw the graph of f.

(b) Find the domain and range of f from the graph.

53. 54.

55–56

■ A function f is given.

(a) Use a graphing calculator to draw the graph of f.

(b) Find the local maximum and minimum values of f and the value of x at which each

occurs.

55. 56.

57–58

■ Find a function that models the quantity described.

57. The number H of hours in d days.

58. The area A of a rectangle whose length is three times its width w.

59–60 ■ Find a formula that models the quantity described.

59. The sum S of the square roots of the numbers , , and .

60. The distance d that a person walking at r feet per second can walk in t minutes.

61–62 ■ Solve the equation to find a formula for the indicated variable.

61. ; for 62. ; for rD = 2x + 2pr√K =

m√

f 1x 2= 3 - 4x

+ 4x

- x

f 1x 2= x

- 3x

+ 1

f 1x 2= 1 - 1x + 2

f 1x 2= 29 - 4x

CONNECTING

THE CONCEPTS

These exercises test your understanding by combining ideas from several sections in a

single problem.

63. Relationship Between Age and Weight The table below gives the ages and weights

of the members of the Hendron family.

John Andrea Helen Louis Ally Victor

Age (years) 46 48 20 16 16 10

Weight (lb) 310 142 120 142 114 90

(a) Find the average age and the median age of the family members.

(b) Find the average weight and the median weight of the family members. Why is

there such a large difference between the average and the median weights? Which

number gives a better description of the central tendency of the Hendrons’ weights?

weight as the output (y). What is the domain of this relation? What is the range?

(d) Make a scatter plot of the relation. Is the relation a function? Why or why not?

(e) Does the scatter plot indicate any general trend in the relationship between age and

weight?

(f) Sketch the line on your scatter plot. Use the graph to determine how many

family members have a weight (in pounds) that is more than five times their age (in

years).

64. Internet Access An Internet provider serves a community action group and charges

its members dollars each month for x minutes of online access time, where f is the

function graphed in the figure.

f 1x 2

y = 5x

500 1000 1500 2000

2500 3000

x 0 100 500 1000 1500 2000 3000

f(x)

$22 $40

120 CHAPTER 1

■

Data, Functions, and Models

(a) Use the graph to complete the following table of values.

(b) What are the domain and range of f ?

(d) On what interval is the function increasing?

(e) What is the minimum monthly charge under this plan? What is the maximum

monthly charge?

CHAPTER 1

■

Review Exercises 121

(a) What is her average score?

(b) What is her median score?

66. Price of Gasoline The price of gasoline dropped rapidly at the Oxxo station in East

Springfield during one week in November 2008. The list gives the price per gallon for

regular unleaded for the first six days that week, rounded to the nearest cent.

CONTEXTS

Score 9 5 6 6 4

Day Sun. Mon. Tue. Wed. Thu. Fri.

Price $2.14 $2.08 $2.02 $1.98 $1.98 $1.92

(a) What was the average price per gallon over this period?

(b) What was the median price per gallon over this period?

median price per gallon for the entire week?

67. Fishing for Salmon A group of tourists hires a boat to fish for salmon in the Strait of

Georgia, British Columbia. The table gives the number of fish each person caught and

the total weight of each person’s catch.

(a) Find the average and the median number of fish caught by each person.

(b) Find the average and the median weight of fish caught by each person.

example, Jack caught 4 fish weighing a total of 18 lb, which gives his fish an

average weight of lb per fish.)

(d) Whose catch had the largest average weight per fish?

(e) What was the average weight per fish for all the fish caught by the entire group?

18>4 = 4.5

(f) Give a verbal description of how the monthly charge is calculated if a customer

uses less than 2000 minutes and if she uses more than 2000 minutes of access time.

(g) Express f algebraically as a piecewise defined function:

(h) Find the net change in the value of f when x changes from 100 to 500 minutes, and

when x changes from 2000 to 3000 minutes.

65. Test Scores The table gives Rianna’s scores on the first five quizzes in her chemistry

class.

f 1x 2= e

_______ if x 6 2000

_______ if x Ú 2000

Person Jack Helen Steve Kalpana Dieter Magda

Number of fish 4 1 2 4 2 2

Weight of fish (lb) 18 4 11 22 13 16

68. Bungee Cord Experiment As part of a science experiment, Qiang attaches a

weight to one end of a strong highly elastic cord whose other end is fixed to a high

platform. He drops the weight and uses a strobe light and a camera to track its fall.

The following table shows the distance between the weight and the platform, at one-

second intervals.

(a) How far has the weight fallen after 2 seconds? After 6 seconds?

(b) What is the approximate time (to the nearest second) that it takes the weight to fall

90 feet?

6 seconds into its fall?

(d) How far does the weight fall between 0 seconds and 1 second? Between 3 seconds

and 4 seconds? Between 5 seconds and 6 seconds?

(e) What pattern or trend do you see in these data?

Time (s) 0 1 2 3 4 5 6 7 8 9

Distance (ft) 0 6.1 23.0 46.5 70.8 90.1 99.5 96.8 82.7 60.5

69. Vending Machines A vending machine operator decides to experiment with the

price she charges for candy bars in her machines, hoping that a higher price might bring

more profits. After several weeks of adjusting her prices, she obtains the results in the

table below. (The cost is calculated on the basis of the fact that each candy bar costs her

25 cents.)

(a) Use the fact that to fill in the last column of the table.

(b) Make a scatter plot of the relationship between price and profit.

charge?

profit = revenue - cost

Price

($)

Number

sold

Revenue

($)

Cost

($)

Profit

($)

0.50 300 150 75

0.70 240 168 60

0.85 220 187 55

1.00 180 180 45

1.25 100 125 25

1.50 60 90 15

70. Air Pressure Declines with Elevation As you rise in elevation above sea level, the

air pressure and the density of the atmosphere decline. This has many consequences for

human beings, including reduced endurance and even difficulty in breathing. (See

Exercise 21 of Section 1.3 on page 33 to see how increasing elevation affects the boiling

point of water.) The following table shows the average atmospheric pressure at various

heights, measured in atmospheres. (1 atm is the air pressure at sea level.)

(a) Make a scatter plot of the data. Do the data points appear to lie on a line?

(b) Use first differences to show that a linear model is appropriate for these data.

(d) Use your model to predict the air pressure at an elevation of 3500 ft.

(e) At what elevation will the air pressure be 0.925 atm?

(f) At an elevation of 20,000 ft, the actual air pressure is about 0.45 atm. How does

this compare with the value predicted by your model? What lesson does this teach

us about using linear models carefully?

122 CHAPTER 1

■

Data, Functions, and Models

CHAPTER 1

■

Review Exercises 123

71. Weight Loss A weight-loss clinic keeps records of its customers’ weight when they

first joined and their weight 12 months later. Some of these data are shown in the

following table.

(a) Graph the relation given by the table.

(b) Use the Vertical Line Test to determine whether the variable y is a function of the

variable x.

Elevation

(ft)

Pressure

(atm)

First

differences

0 1.00 —

500 0.98

1000 0.96

1500 0.94

2000 0.92

2500 0.90

x ⴝ Beginning weight

212 161 165 142 170 181 165 172 312 302

y ⴝ Weight after 12 months

188 151 166 131 165 156 156 152 276 289

72. Bantam Eggs A hobby farmer maintains a small flock of bantam hens, whose eggs

he sells to a local health food store. He is interested in finding how the hens’ egg-laying

capacity is related to their age, so he gathers the data shown in the graph in the margin.

The scatter plot relates the age of each hen (in years) to the number of eggs she laid in

one week.

(a) Make a table of the ordered pairs in the relation given by the graph.

(b) List the ordered pair(s) with input 2 and the ordered pair(s) with input 4.

(d) Use the Vertical Line Test to determine whether the relation is a function.

73. Sales Commission André sells office furniture and earns a commission based on his

sales. The piecewise defined function C gives his commission as a function of his

monthly sales x (in dollars):

(a) Find C(1000), C(4000), and C(20,000). What do these values represent?

(b) Sketch a graph of C. Use your graph to determine whether it is possible for André

to earn a commission of $1250. Why or why not?

(d) Find the net change in André’s commission if his sales increase from $9000 to

$14,000 per month.

74. Cooking a Roast Jason takes a roast beef out of the freezer before going to work in

the morning, leaving it to defrost on the kitchen counter. He cooks it for two hours in

the oven, then lets it rest for 30 minutes before serving it to his dinner guests. Draw a

rough graph of the changes in the temperature of the roast over the course of the day.

C1x 2= c0.1x

 if 3000 … x 6 10,000

0  if 0 … x 6 3000

0.15xif x Ú 10,000

Eggs

1234

Age (yr)