Beginning and Intermediate Algebra_3e_Ch08(1)

7/22/2019 Beginning and Intermediate Algebra_3e_Ch08(1)

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502

Our competitive

advantage is our math

skills, which is probably

not something you

would expect of a

media company.

MAX LEVCHIN,

CEO OF SLIDE(Slide is the number one company

for writing Facebook applications.)

Every day millions of people create trillions of bytes of information. The only waywe can make sense out of these data and determine what is occurring within society isto use mathematics. One of the most important mathematical concepts used to discovertrends and patterns is that of afunction. A function typically receives an input (or ques-tion), performs a computation, and gives the output (or answer).

Functions have been used in science and engineering for centuries to answer ques-tions related to things like eclipses, communication, and transportation. However, todayfunctions are also being used to describe human behavior and to design social networks.(See Section 8.1, Exercise 75.) In fact, you may have noticed that new features availableon Twitter and Facebook are sometimes referred to as applications or functions. Peopleare creating thousands of new functions every day.Math skills are essential for writingsuccessful applications and functions.

8.1 Functions and Their

Representations8.2 Linear Functions8.3 Compound Inequalities8.4 Other Functions and

Their Properties8.5 Absolute Value

Equations andInequalities

Introduction to Functions8

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley.Copyright 2013 by Pearson Education, Inc.


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5038.1 FUNCTIONS AND THEIR REPRESENTATIONS

8.1 Functions and Their Representations

Basic Concepts Representations of a Function Definition of a Function Identifying a Function Graphing Calculators (Optional)

A LOOK INTO MATH N

In earlier chapters we showed how to use numbers to describe data. For example, insteadof simply saying that there are a lotof people on Twitter, we might say that there are abou50 million tweets per day. A number helps explain what a lot means. We also showedthat data can be summarized with formulas and graphs. Formulas and graphs are some-times used to represent functions, which are essential in mathematics. In this section weintroduce functions and their representations.

Basic ConceptsN REAL-WORLD CONNECTION Functions are used to calculate many important quantities. For

example, suppose that a person works for $7 per hour. Then we could use a function namedfto calculate the amount of money the person earned after workingxhours simply by mul-tiplying the input xby 7. The resulty is called the output. This concept is shown visually inthe following diagram.

Function f

Uses inputx tocompute a value ofy

Outputy =f(x)Inputx

For each valid inputx, a function computes exactly one outputy, which may be repre-sented by the ordered pair (x,y). If the input is 5 hours,foutputs 7 # 5 = $35; if the inpuis 8 hours, foutputs 7 # 8 = $56. These results can be represented by the ordered pairs(5, 35) and (8, 56). Sometimes an input may not be valid. For example, ifx= - 3, there is

no reasonable output because a person cannot work - 3 hours.We say thaty is a function of xbecause the outputy is determined by and depends on

the inputx. As a result,y is called the dependent variable andxis the independent variableTo emphasize thaty is a function ofx, we use the notation y = f(x). The symbol f(x) doesnot represent multiplication of a variablefand a variablex. The notation y = f(x) is called

function notation, is read y equalsfofx, and means that functionfwith inputxproduce

outputy. For example, ifx= 3 hours,y = f(3) = $21.

NEW VOCABULARYn Functionn Function notationn Input/Outputn Name of the functionn Dependent variablen Independent variablen Verbal representationn Numerical representationn Symbolic representationn Graphical representationn Diagrams/Diagrammatic

representation

n Relationn Domain/Rangen Nonlinear functionsn Vertical line test

FUNCTION NOTATION

The notation y = f(x) is called function notation. The input isx, the output isy,and the name of the function isf.

Name

Q

y = f(x)

Q

Q

Output Input

The variabley is called the dependent variable and the variablexis called the inde-pendent variable. The expression f(4) = 28 is read fof 4 equals 28 and indicatesthatfoutputs 28 when the input is 4. A function computes exactly one output for eachvalid input. The lettersf, g, and h are often used to denote names of functions.

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504 CHAPTER 8 INTRODUCTION TO FUNCTIONS

NOTE: Functions can be given meaningful names and variables. For example, functionfcould have been defined by P(h) = 7h, where function P calculates the pay after workingh hours for $7 per hour.

N REAL-WORLD CONNECTION Functions can be used to compute a variety of quantities. Forexample, suppose that a boy has a sister who is exactly 5 years older than he is. If the ageof the boy isx, then a function g can calculate the age of his sister by adding 5 to x. Thus

g(4) = 4 + 5 = 9, g(10) = 10 + 5 = 15, and in general g(x) = x + 5. That is, func-tion g adds 5 to inputxto obtain the outputy = g(x).

Functions can be represented by an inputoutput machine, as illustrated in Figure 8.1.This machine represents function g and receives inputx= 4, adds 5 to this value, and thenoutputs g(4) = 4 + 5 = 9.

Representations of a FunctionN REAL-WORLD CONNECTION A functionfforms a relation between inputsxand outputsy that

can be represented verbally, numerically, symbolically, and graphically. Functions can also berepresented with diagrams. We begin by considering a functionfthat converts yards to feet.

VERBAL REPRESENTATION (WORDS) To convertxyards toy feet we multiplyxby 3.Therefore, if functionfcomputes the number of feet inxyards, a verbal representationoffis Multiply the inputxin yards by 3 to obtain the outputy in feet.

NUMERICAL REPRESENTATION (TABLE OF VALUES) A function f that convertsyards to feet is shown in Table 8.1, wherey = f(x).

A table of values is called a numerical representation of a function. Many times it is impos-sible to list all valid inputsxin a table. On the one hand, if a table does not contain everyx-input,it is apartial numerical representation. On the other hand, a complete numerical representationincludes all valid inputs. Table 8.1 is a partial numerical representation offbecause many valid

inputs, such asx= 10 orx= 5.3, are not shown in it. Note that for each valid inputxthere isexactly one outputy. For a function, inputs are not listed more than once in a table.

SYMBOLIC REPRESENTATION (FORMULA) Aformula provides a symbolic repre-sentation of a function. The computation performed by f to convert xyards to y feet isexpressed by y = 3x. A formula forf is f(x) = 3x, where y = f(x). We say that function

f is defined by or given byf(x) = 3x. Thus f(2) = 3 # 2 = 6.GRAPHICAL REPRESENTATION (GRAPH) A graphical representation, or graph,visually associates anx-input with ay-output. The ordered pairs

(1, 3), (2, 6), (3, 9), (4, 12), (5, 15), (6, 18), and (7, 21)

Figure 8.1

4 (Inputx)

9 =g(4)(Outputy)

Inputx

Add 5

4

4 + 5

Output g(x)

Function Machine

for g(x) x 5

TABLE 8.1

x(yards) y (feet)

1 3

2 6

3 9

4 12

5 15

6 18

7 21



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from Table 8.1 are plotted in Figure 8.2(a). This scatterplot suggests a line for the graph fFor each real number x there is exactly one real number y determined by y = 3x. If werestrict inputs to x 0 and plot all ordered pairs (x, 3x), then a line with no breaks wilappear, as shown in Figure 8.2(b).

Because f(1) = 3, it follows that the point (1,3) lies on the graph off, as shown inFigure 8.3. Graphs can sometimes be used to define a functionf. For example, because thepoint (1,3) lies on the graph offin Figure 8.3, we can conclude that f(1) = 3. That is, eachpoint on the graph offdefines an inputoutput pair forf.

Figure 8.2

40 8 12 16 20 24

4

8

12

16

2024

x

y

(1, 3)

(2, 6)

(3, 9)

(4, 12)

(5, 15)

(6, 18)

(7, 21)

Yards

(a)

Feet

Yards

(b)

Feet

40 8 12 16 20 24

4

8

12

16

20

24

x

y

y = 3x

Yards to Feet

Figure 8.3

10

f(1) = 3

x

y

(1,3)

y =f(x)

Evaluating f(1)

MAKING CONNECTIONS

Functions, Points, and Graphs

Iff(a) = b, then the point (a,b) lies on the graph off. Conversely, if the point (a,b) lieson the graph off, then f(a) = b. See Figure 8.4(a). Thus each point on the graph offcan bewritten in the form (a,f(a)). See Figure 8.4(b).

Function fDefined by a Graph

Figure 8.4

a

b

y = f(x)

x

y

(a)

(a, b)

f(a) = b

a

f(a)

x

y

(b)

(a,f(a))

Points on Graph off

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DIAGRAMMATIC REPRESENTATION (DIAGRAM) Functions may be represented bydiagrams. Figure 8.5 is a diagram of a function where an arrow is used to identify theoutputy associated with inputx. For example, an arrow is drawn from input 2 to output 6,which is written in function notation as f(2) = 6. That is, 2 yards are equivalent to 6 feet.

Figure 8.6(a) shows a (different) function f even though f(1) = 4 and f(2) = 4.Although two inputs for fhave the same output, each valid input has exactly one output.In contrast, Figure 8.6(b) is nota function because input 2 results in two different outputs,

5 and 6.

Figure 8.6

1

2

3

4

5

f

(a) Function

1

2

4

5

6

(b) Not a Function

MAKING CONNECTIONSFour Representations of a Function

Symbolic Representation f(x) = x + 1Numerical Representation Graphical Representation

Verbal Representation fadds 1 to an inputxto produce an outputy.

x y

- 2 - 1

- 1 0

0 1

1 22 3

3 2 1 2 3

3

2

1

1

2

3

x

y

y =x + 1

STUDY TIP

Be sure that you understandwhat verbal, numerical,graphical, and symbolicrepresentations are.

EXAMPLE 1 Evaluating symbolic representations (formulas)

Evaluate each functionfat the given value ofx.(a) f(x) = 3x - 7 x= -2(b) f(x) = x

x + 2 x= 0.5

(c) f(x) = 2x - 1 x= 10Solution(a) f( 2) = 3( 2) - 7 = -6 - 7 = 13(b) f(0.5) = 0.5

0.5 + 2=

0.5

2.5= 0.2

(c) f(10) = 210 - 1 = 29 = 3Now Try Exercises 21, 23, 31

1

2

3

3

6

9

f

Function

Figure 8.5

Yards to Feet



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EXAMPLE 2 Calculating sales tax

Let a functionfcompute a sales tax of 7% on a purchase ofxdollars. Use the given representation to evaluate f(2).(a) Verbal Representation Multiply a purchase ofxdollars by 0.07 to obtain a sales tax of

y dollars.(b) Numerical Representation(partial) Shown in Table 8.2(c) Symbolic Representation f(x) = 0.07x(d) Graphical Representation Shown in Figure 8.7(e) Diagrammatic Representation Shown in Figure 8.8

Solution(a) Verbal Multiply the input 2 by 0.07 to obtain 0.14. The sales tax on a $2.00 purchase

is $0.14.(b) Numerical From Table 8.2, f(2) = $0.14.(c) Symbolic Because f(x) = 0.07x, f(2) = 0.07(2) = 0.14, or $0.14.(d) Graphical To evaluate f(2) with a graph, first find 2 on thex-axis in Figure 8.9. Then

move vertically upward until you reach the graph off. The point on the graph may beestimated as (2, 0.14), meaning that f(2) = 0.14. Note that it may not be possible tofind the exact answer from a graph. For example, one might estimate f(2) to be 0.13 or0.15 instead of 0.14.

Figure 8.9

0 1 2 3 4 5 6

0.10.14Output

Input

0.2

0.3

0.4

x

y

(2, 0.14)

f(2) = 0.14

Evaluating a Function

N REAL-WORLD CONNECTION In the next example we calculate sales tax by evaluating differentrepresentations of a function.

(e) Diagrammatic In Figure 8.8, follow the arrow from 2 to 0.14. Thus f(2) = 0.14.Now Try Exercises 25, 33, 53, 59, 61

TABLE 8.2

x f(x)

$1.00 $0.07

$2.00 $0.14

$3.00 $0.21

$4.00 $0.28

Figure 8.7

10 2 3 4 5 6

0.1

0.2

0.3

0.4

x

y

Purchase (dollars)

SalesTax(dollars)

Figure 8.8

1

2

3

4

0.07

0.14

0.21

0.28

f

Sales Tax of 7%

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EXAMPLE 3 Computing crutch length

People who sustain leg injuries often require crutches. A proper crutch length can be esti-mated without using trial and error. The functionL, given byL(t) = 0.72t + 2, outputs anappropriate crutch length L in inches for a person tinches tall. (Source:Journal of the American Physi-

cal Therapy Association.)

(a) FindL(60) and interpret the result.(b) If one person is 70 inches tall and another person is 71 inches tall, what should be the

difference in their crutch lengths?

Solution(a) L(60) = 0.72(60) + 2 = 45.2. Thus a person 60 inches tall needs crutches that are

about 45.2 inches long.(b) From the formulaL(t) = 0.72t + 2, we can see that each 1-inch increase in tresults in

a 0.72-inch increase inL(t). For example,

L(71) - L(70) = 53.12 - 52.4 = 0.72.

Now Try Exercise 75

In the next example we find a formula and then sketch a graph of a function.

EXAMPLE 4 Finding representations of a function

Let functionfsquare the inputxand then subtract 1 to obtain the outputy.(a) Write a formula, or symbolic representation, forf.(b) Make a table of values, or numerical representation, forf. Usex= -2, - 1, 0, 1, 2.

(c) Sketch a graph, or graphical representation, off.

Solution(a) Symbolic Representation If we squarexand then subtract 1, we obtainx2 - 1. Thus a

formula forfis f(x) = x2 - 1.(b) Numerical Representation Make a table of values for f(x), as shown in Table 8.3. For

example,

f( 2) = ( 2)2 - 1 = 4 - 1 = 3.(c) Graphical Representation To obtain a graph of f(x) = x2 - 1, plot the points from

Table 8.3 and then connect them with a smooth curve, as shown in Figure 8.10. Note thatwe need to plot enough points so that we can determine the overall shape of the graph.

t

L

READING CHECK

Give a verbal, numerical,symbolic, and graphicalrepresentation of a functionthat calculates the numberof days in a given numberof weeks. Choose meaning-ful variables.

TABLE 8.3

x f(x) 2 3

- 1 0

0 -1

1 0

2 3Figure 8.10

3 2 1 32

3

2

1

2

3

x

y

y =x2 1

Now Try Exercise 63

N REAL-WORLD CONNECTION There are many examples of functions. To give more meaningto a function, sometimes we change both its name and its input variable. For instance, if weknow the radius rof a circle, we can calculate its circumference by using C(r) = 2pr. Thenext example illustrates how functions are used in physical therapy.



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Definition of a FunctionA function is a fundamental concept in mathematics. Its definition should allow for all rep-resentations of a function.A function receives an input x and produces exactly one output ywhich can be expressed as an ordered pair:

(x,y).

Q

Q

Input Output

A relation is a set of ordered pairs, and a function is a special type of relation.

FUNCTION

A functionfis a set of ordered pairs (x,y) where eachx-value corresponds to exactlyoney-value.

The domain offis the set of all x-values, and the range offis the set of ally-valuesFor example, a functionfthat converts 1, 2, 3, and 4 yards to feet could be expressed as

f= {(1, 3), (2, 6), (3, 9), (4, 12)}.

The domain offisD = {1,2,3,4}, and the range offisR = {3,6,9,12}.

MAKING CONNECTIONS

Relations and Functions

A relation can be thought of as a set of inputoutput pairs. A function is a special type ofrelation whereby each input results in exactly one output.

N REAL-WORLD CONNECTION In the next example, we see how education can improve a persons chances for earning a higher income.

EXAMPLE 5 Computing average income

A function fcomputes the average individual income in dollars in relation to educationaattainment. This function is defined by f(N) = 21,484, f(H) = 31,286, f(B) = 57,181and f(M) = 70,181, whereNdenotes no diploma,Ha high school diploma,B a bachelorsdegree, andMa masters degree. (Source: 2010 Statistical Abstract.)(a) Writefas a set of ordered pairs.(b) Give the domain and range off.(c) Discuss the relationship between education and income.

Solution(a) f= {(N, 21484), (H, 31286), (B, 57181), (M, 70181)}(b) The domain of functionfis given by D = {N,H,B,M}, and the range of functionfis

given byR = {21484, 31286, 57181, 70181}.(c) Education paysthe greater the educational attainment, the greater are annual earnings

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EXAMPLE 6 Finding the domain and range graphically

Use the graphs offshown in Figures 8.11 and 8.12 to find each functions domain andrange.(a) (b)

Figure 8.11

3 2 1 1 2 3

3

2

1

1

2

x

y

y = f(x)

Figure 8.12

4 2 1 2 3 4

4

3

2

1

1

4

x

y

y = f(x)

Solution(a) The domain is the set of all x-values that correspond to points on the graph off.

Figure 8.13 shows that the domain D includes allx-values satisfying 3 " x " 3.(Recall that the symbol is read less than or equal to.) Because the graph is a semi-

circle with no breaks, the domain includes all real numbers between and including -3and 3. The range R is the set ofy-values that correspond to points on the graph off.ThusR includes ally-values satisfying 0 " y " 3.

Figure 8.13

3 2 1 1 2 3

3

2

1

1

2

x

y

Domain

Range

Figure 8.14

4 2 1 2 3 4

3

2

1

1

4

x

y

Range

Domain

(b) The arrows on the ends of the graph in Figure 8.12 indicate that the graph extends

indefinitely left and right, as well as upward. Thus D includes all real numbers.See Figure 8.14. The smallesty-value on the graph is y = - 3, which occurs whenx= 2. Thus the range R is y # 3. (Recall that the symbol is read greaterthan or equal to.)

Now Try Exercises 77, 81

CRITICAL THINKING

Suppose that a car travels at 50 miles per hour to a city that is 250 miles away. Sketch a graph ofa function f that gives the distanceytraveled afterxhours. Identify the domain and range off.

The domain of a function is the set of all valid inputs. To determine the domain of a func-

tion from a formula, we must find x-values for which the formula is defined. To do this, wemust determine if we can substitute any real number in the formula for f(x). If we can, thenthe domain offis all real numbers. However, there are situations in which we must limit thedomain off. For example, the domain must often be limited when there is either division or asquare root in the formula forf. When division occurs, we must be careful to avoid values ofthe variable that result in division by 0, which is undefined. When a square root occurs, we

READING CHECK

Use the graph inFigure 8.12 to evaluate f(3).



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Identifying a Function

Recall that for a function each valid inputxproduces exactly one outputy. In the next threeexamples we demonstrate techniques for identifying a function.

EXAMPLE 8 Determining whether a set of ordered pairs is a function

The set Sof ordered pairs (x,y) represents the number of mergers and acquisitions y in2010 for selected technology companiesx.

S= 5(IBM,12), (HP,7), (Oracle,5), (Apple,5), (Microsoft,0)6Determine ifSis a function. (Source: cbinsights.)

Figure 8.15 D: all real numbersR:y 0

1 22 1

2

1

1

2

x

y

y = |x|

(a)

x

y

-3 -2 1 2 3-1

2

3

4

5

y =x 2

(b)

D: all real numbers

R: y 0

x

y

2 4 6 8

2

4

6

8

y = x

(c)

D:x 0

R:y 0

Absolute value: f(x) x x -2 - 1 0 1 2

x 2 1 0 1 2

Square: f(x) x2 x -2 - 1 0 1 2

x2 4 1 0 1 4

Square root: f(x) 1x x 0 1 4 9

1x 0 1 2 3

must be careful to avoid values of the variable that result in the square root of a negative num-ber, which is not a real number. This concept is demonstrated in the next example.

EXAMPLE 7 Finding the domain of a function

Use f(x) to find the domain off.

(a) f(x) = 5x (b) f(x) = 1x - 2 (c) f(x) = 1xSolution(a) Because we can always multiply a real numberxby 5,f(x) = 5xis defined for all real

numbers. Thus the domain offincludes all real numbers.(b) Because we cannot divide by 0, input x= 2 is not valid for f(x) = 1x - 2. The expres-

sion for f(x) is defined for all other values ofx. Thus the domain offincludes all reanumbers except 2, orx 2.

(c) Because square roots of negative numbers are not real numbers, the inputs forf(x) = 1x cannot be negative. Thus the domain off includes all nonnegative num-bers, orx 0.

Now Try Exercises 87, 91, 95

Symbolic, numerical, and graphical representations of three common functions areshown in Figure 8.15. Note that their graphs are not lines. For this reason they are callednonlinear functions. Use the graphs to find the domain and range of each function.

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SolutionThe input x is the name of the technology company, and the output y is the number of

mergers and acquisitions associated with that company. The set S is a function becauseeach company x is associated with exactly one number y. Note that even though therewere 5 mergers and acquisitions corresonding to both Oracle and Apple, Sis nonethelessa function.

Now Try Exercise 123

EXAMPLE 9 Determining whether a table of values represents a function

Determine whether Table 8.4 represents a function.

SolutionThe table does not represent a function because inputx= 1 produces two outputs: - 4 and 5.That is, the following two ordered pairs both belong to this relation.

Same inputx

Q Q

(1, 4) (1, 5) Q

Not a function

Q

Q

Different outputsy


VERTICAL LINE TEST To determine whether a graph represents a function, we must beconvinced that it is impossible for an inputxto have two or more outputsy. If two distinctpoints have the samex-coordinate on a graph, then the graph cannot represent a function. Forexample, the ordered pairs ( 1, 1) and ( 1, -1) could not lie on the graph of a functionbecause input 1 results in two outputs: 1 and - 1. When the points ( -1, 1) and ( - 1, -1)are plotted, they lie on the same vertical line, as shown in Figure 8.16(a). A graph passingthrough these points intersects the vertical line twice, as illustrated in Figure 8.16(b).

TABLE 8.4

x y

1 -4

2 8

3 2

1 5

4 -6

Figure 8.16

1 2

2

1

1

2

x

y

(1, 1)

(1, 1)

(a)

1 2

2

1

1

2

x

y

(1, 1)

(1, 1)

(b)

Two or morepoints lie on thesame vertical line:nota function

To determine whether a graph represents a function, visualize vertical lines moving

across thexy-plane. If each vertical line intersects the graph at most once, then it is a graphof a function. This test is called the vertical line test. Note that the graph in Figure 8.16(b)fails the vertical line test and therefore does not represent a function.

READING CHECK

What is the vertical line testused for?

VERTICAL LINE TEST

If every vertical line intersects a graph at no more than one point, then the graphrepresents a function.



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EXAMPLE 10 Determining whether a graph represents a function

Determine whether the graphs shown in Figure 8.17 represent functions.

Figure 8.17

4 4

4

2

4

x

y

(a)

x

y

2

1

12

3 1 3

(b)

x

y

2

1

2

1

2 1 1 2

(c)

Solution(a) Visualize vertical lines moving across thex y-plane from left to right. Any (red) vertica

line will intersect the graph at most once, as depicted in Figure 8.18(a). Therefore thegraph does represent a function.

(b) Visualize vertical lines moving across the x y-plane from left to right. The graph doesnotrepresent a function because there exist (red) vertical lines that can intersect thegraph twice. One such line is shown in Figure 8.18(b).

(c) Visualize vertical lines moving across the x y-plane from left to right. The graph is ascatterplot and does notrepresent a function because there exists one (red) vertical linethat intersects two points: (1, 1) and (1, -2) with the same x-coordinate, as shown inFigure 8.18(c).


Graphing Calculators (Optional)Graphing calculators provide several features beyond those found on scientific calculatorsGraphing calculators have additional keys that can be used to create tables, scatterplots, andgraphs.

N REAL-WORLD CONNECTION The viewing rectangle, or window, on a graphing calculatoris similar to the viewfinder in a camera. A camera cannot take a picture of an entire scene

Figure 8.18

4 4

4

2

4

x

y

(a)

Passes Vertical Line Test

x

y

2

1

1

2

3 1 3

(b)

Fails Vertical Line Test

x

y

2

1

2

1

2 1 2

(c)

Fails Vertical Line Test

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The camera must be centered on some object and can photograph only a portion of theavailable scenery. A camera can capture different views of the same scene by zoomingin and out, as can graphing calculators. The xy-plane is infinite, but the calculator screencan show only a finite, rectangular region of thexy-plane. The viewing rectangle must bespecified by setting minimum and maximum values for both the x- and y-axes before agraph can be drawn.

We use the following terminology regarding the size of a viewing rectangle. Xminis the minimumx-value along thex-axis, and Xmax is the maximumx-value. Similarly,Ymin is the minimumy-value along they-axis, and Ymax is the maximumy-value. Most

graphs show anx-scale and ay-scale with tick marks on the respective axes. Sometimesthe distance between consecutive tick marks is 1 unit, but at other times it might be 5 or10 units. The distance represented by consecutive tick marks on the x-axis is called Xscl,

and the distance represented by consecutive tick marks on they-axis is called Yscl (seeFigure 8.19).

This information about the viewing rectangle can be written as [Xmin, Xmax, Xscl] by[Ymin, Ymax, Yscl]. For example, [ -10, 10, 1] by [ -10, 10, 1] means that Xmin = - 10,Xmax = 10, Xscl = 1, Ymin = -10, Ymax = 10, and Yscl = 1. This setting isreferred to as the standard viewing rectangle. The window in Figure 8.19 is [- 3, 3, 1] by[ -3, 3, 1].

EXAMPLE 11 Setting the viewing rectangle

Show the viewing rectangle [ -2, 3, 0.5] by [ -100, 200, 50] on your calculator.

SolutionThe window setting and viewing rectangle are displayed in Figure 8.20. Note that inFigure 8.20(b) there are 6 tick marks on the positivex-axis because its length is 3 units andthe distance between consecutive tick marks is 0.5 unit.


SCATTERPLOTS Many graphing calculators have the capability to create scatterplotsand line graphs. The next example illustrates how to make a scatterplot with a graphingcalculator.

Figure 8.19

XmaxXmin

Yscl

Xscl

Ymax

Ymin

CALCULATOR HELP

To set a viewing rectangle, see

Appendix A (page AP-3).

Figure 8.20

WINDOWXm i n2Xmax3X s c l.5Ym i n100Ymax200Y s c l50X r e s1

(a) (b)

[-2, 3, 0.5] by [ - 100, 200, 50]



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CALCULATOR HELP

To make a scatterplot,

see Appendix A

(pages AP-3 and AP-4).

Figure 8.21

L2(5)

L1 L2 L3 2

(a)

2 2 ---- --1 31 22 3------ ------

Plot1 P lo t2 P lo t3On OffType:

X l i s t : L1Y l i s t : L2Mark:

(b)

+

(c)

[-4, 4, 1] by [ -4, 4, 1]

EXAMPLE 12 Making a scatterplot with a graphing calculator

Plot the points (- 2, -2), (- 1, 3), (1, 2), and (2, - 3) in [ - 4, 4, 1] by [ - 4, 4, 1].

SolutionWe entered the points ( -2, - 2), ( -1, 3), (1, 2), and (2, - 3) shown in Figure 8.21(a), usingthe STAT EDIT feature. The variable L1 represents the list ofx-values, and the variable L2

represents the list ofy-values. In Figure 8.21(b) we set the graphing calculator to make ascatterplot with the STATPLOT feature, and in Figure 8.21(c) the points have been plottedIf you have a different model of calculator you may need to consult your owners manual.


GRAPHS AND TABLES We can use graphing calculators to create graphs and tables, usu-ally more efficiently and reliably than with pencil-and-paper techniques. However, a graph-ing calculator uses the same techniques that we might use to sketch a graph. For example,one way to sketch a graph ofy = 2x - 1 is first to make a table of values, as shown in

Table 8.5.

We can plot these points in thex y-plane, as shown in Figure 8.22. Next we might con-nect the points, as shown in Figure 8.23.

In a similar manner, a graphing calculator plots numerous points and connects them to

make a graph. To create a similar graph with a graphing calculator, we enter the formulaY1 = 2X - 1, set an appropriate viewing rectangle, and graph as shown in Figures 8.24and 8.25. A table of values can also be generated as illustrated in Figure 8.26.

CALCULATOR HELP

To make a graph, see Appendix A

(page AP-5). To make a table, see

Appendix A (pages AP-2 and AP-3).

TABLE 8.5x y

- 1 -3

0 -1

1 1

2 3

Figure 8.23

3 2 1 1 2 3

3

1

1

2

3

x

y

y = 2x 1

Graphing a Line

Figure 8.22

3 2 1 1 2 3

2

1

1

2

3

x

y

Plotting Points

Figure 8.24

Plot1 Plot2 Plot3\Y12X1\Y2\Y3\Y4\Y5\Y6\Y7

Figure 8.25

y = 2x 1

Figure 8.26

Y12X1

X Y11 30 11 12 33 54 75 9

[-10, 10, 1] by [ -10, 10, 1]

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8.1 Putting It All Together

A function calculates exactly one output for each valid input and produces inputoutputordered pairs in the form (x, y). A function typically computes something such as area,speed, or sales tax.

A function can be represented verbally, symbolically, numerically, and graphically.

Verbal Precise word description of what iscomputed

May be oral or writtenMust be statedprecisely

Symbolic Mathematical formula Efficient and concise way of represent-ing a function (e.g., f(x) = 2x - 3)

Numerical List of specific inputs and theiroutputs

May be in the form of a table or anexplicit set of ordered pairs

REPRESENTATION EXPLANATION COMMENTS

Graphical, diagrammatic Shows inputs and outputsvisually

No words, formulas, or tablesMany types of graphs and diagrams arepossible.

Function A set of ordered pairs (x,y), whereeachx-value corresponds to exactlyoney-value

f= {(1, 3), (2, 3), (3, 1)}f(x) = 2xA graph ofy = x + 2A table of values for y = 4x

Independent Variable The inputvariable for a function Function Independent Variablef(x) = 2x xA(r) = pr2 rV(s) = s3 s

CONCEPT EXPLANATION EXAMPLES

Dependent Variable The outputvariable of a function

There is exactly one output for eachvalid input.

Function Dependent Variable

y = f(x) yT= F(r) TV= g(r) V

Domain and Rangeof a Function

The domainD is the set of all validinputs. The rangeR is the set of alloutputs.

For S= {(- 1, 0), (3, 4), (5, 0)},D = { -1, 3, 5} andR = {0, 4}.For f(x) = 1x the domain includes allreal numbers except 0, orx 0.

Vertical Line Test If every vertical line intersects a graphat no more than one point, the graphrepresents a function.

This graph does notpass this test andthus does not represent a function.

x

y

Two points lie on thesame vertical line:nota function



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CONCEPTS AND VOCABULARY

23. f(x) = 1x x= 0, 94 0; 3224. f(x) = 23 x x= -1, 27 -1; 325. f(x) = x2 x= -5, 32 25; 9426. f(x) = x3 x= -2, 0.1 - 8; 0.00127. f(x) = 3 x= -8, 73 3; 328. f(x) = 100 x= -p, 13 100; 10029. f(x) = 5 - x3 x= -2, 3 13; -2230. f(x) = x2 + 5 x= - 12, 6 214 ; 4131. f(x) = 2

x + 1 x= -5, 4 - 12;

25

32. f(x) = xx - 4

x= -3, 1 37; -13

Exercises 3338: Do the following.

(a) Write a formula for the function described.

(b) Evaluate the function for input10 and interprethe result.*

33. FunctionIcomputes the number of inches inxyards(a)I(x) = 36x (b)I(10) = 360

34. FunctionA computes the area of a circle with radius r(a) A(r) = pr2 (b) A(10) = 100p 314.2

35. FunctionMcomputes the number of miles inxfeet.(a)M(x) = x5280 (b)M(10) =

105280 0.0019

36. Function C computes the circumference of a circlewith radius r. (a) C(r) = 2pr (b) C(10) = 20p 62.8

37. FunctionA computes the square feet inxacres. (HintThere are 43,560 square feet in one acre.)(a) A(x) = 43,560x (b) A(10) = 435,600

38. Function Kcomputes the number of kilograms in xpounds. (Hint: There are about 2.2 pounds in onekilogram.) (a) K(x) = x2.2 (b) K(10) =

102.2 = 4.54

Exercises 3942: Write each function f as a set of ordered

pairs. Give the domain and range of f.

39. f(1) = 3, f(2) = - 4, f(3) = 0f= {(1, 3), (2, -4), (3, 0)};D = {1, 2, 3},R = {-4, 0, 3}

40. f( -1) = 4, f(0) = 6, f(1) = 4f= {( - 1, 4), (0, 6), (1, 4)};D = { -1, 0, 1},R = {4, 6}

41. f(a) = b, f(c) = d, f(e) = a, f(d) = bf= {(a, b), (c, d), (e, a), (d, b)};D = {a, c, d, e},R = {a, b, d}

42. f(a) = 7, f(b) = 7, f(c) = 7, f(d) = 7f= {(a, 7), (b, 7), (c, 7), (d, 7)};D = {a, b, c, d},R = {7}

Exercises 4352: Sketch a graph of f.*

43. f(x) = -x + 3 44. f(x) = -2x + 145. f(x) = 2x 46. f(x) = 12x - 2

1. The notationy = f(x) is called notation. function

2. The notationy = f(x) is read .y equalsfofx3. The notation f(x) = x2 + 1 is a(n) representa-

tion of a function. symbolic

4. A table of values is a(n) representation of afunction. numerical

5. The set of valid inputs for a function is the . domain

6. The set of outputs for a function is the range .

7. A function computes one output for each valid input.

8. (True or False?) The vertical line test is used to iden-tify graphs of relations. F

9. (True or False?) Four ways to represent functions are

verbal, numerical, symbolic, and graphical. T

10. Iff(3) = 4, the point (3, 4) is on the graph off. If(3, 6) is on the graph off, thenf( 3 ) = 6 .

11. Thinking Generally Iff(a) = b, the point (a, b) ison the graph off.

12. Thinking Generally If (c, d) is on the graph of g,then g(c) = d .

13. Thinking Generally Ifa is in the domain off, then

f(a) represents how many outputs? 114. Thinking Generally Iff(x) = x for every x in the

domain off, then the domain and range offare equal .

Exercises 1520: Determine whether the phrase describes

a function.

15. Calculating the square of a number Yes

16. Calculating the low temperature for a day Yes

17. Listing the students who passed a given math exam No

18. Listing the children of parentx No

19. Finding sales tax on a purchase Yes

20. Naming the people in your class No

REPRESENTING AND EVALUATING FUNCTIONS

Exercises 2132: Evaluate f(x) at the given values of x.21. f(x) = 4x - 2 x= -1, 0 -6; -222. f(x) = 5 - 3x x= - 4, 2 17; -1

8.1 Exercises

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CHAPTER 8 INTRODUCTION TO FUNCTIONS518

47. f(x) = 4 - x 48. f(x) = 6 - 3x49. f(x) = x2 50. f(x) = 1x51. f(x) = 1x + 1 52. f(x) = 12x2 - 1

Exercises 5358: Use the graph of f to evaluate the given

expressions.

53. f(0) and f(2) 3; -1 54. f( -2) and f(2) -2; 0

x

y

-3

-2

-1

1

2

3

-3 -2 -1 1 3

x

y

-2

1

2

-2 -1 2

55. f( -2) and f(1) 0; 2 56. f( -1) and f(0) 1; 0

x

y

-3 -1 1 3

-3

1

2

3

x

y

-2

-1

1

2

-2 -1 1 2

57. f(1) and f(2) -4; -3 58. f( -1) and f(4) 3; 2

x

y

-4 -2 1 2 4

-4

-3

1

2

3

4

x

y

-1 1 2 3 4 5-1

1

3

4

5

Exercises 59 and 60: Use the table to evaluate the given

expressions.

59. f(0) and f(2) 5.5; 3.7

60. f( -10) and f(5) 23; -33

Exercises 61 and 62: Use the diagram to evaluate f(1990).Interpret your answer.

61. The function f computes average fuel efficiency ofnew U.S. passenger cars in miles per gallon duringyearx. (Source: Department of Transportation.) 26.9

1970

1980

1990

2000

2010

14.1

22.6

26.9

28.5

32.6

f

62. The function f computes average cost of tuition atpublic colleges and universities during academic yearx.(Source: The College Board.)$1809

804

1809

3510

7020

f

1980

1990

2000

2010

x 0 1 2 3 4

f(x) 5.5 4.3 3.7 2.5 1.9

x -10 - 5 0 5 10

f(x) 23 96 - 45 - 33 23

In 1990 average

fuel efficiency

was 26.9 mpg.

In 1990, the averagecost of tuition was$1809.

Exercises 6366: Express the verbal representation for

the function f numerically, symbolically, and graphically.

Let x= - 3, -2, -1, p , 3 for the numerical represen-tation (table), and let -3 x 3 for the graph.*

63. Add 5 to the inputxto obtain the outputy. y = x + 5

64. Square the inputxto obtain the outputy.y = x2

65. Multiply the inputxby 5 and then subtract 2 to obtainthe outputy. y = 5x - 2

66. Divide the inputxby 2 and then add 3 to obtain theoutputy. y = x2 + 3

Exercises 6772: Give a verbal representation for f(x).*67. f(x) = x - 12 68. f(x) = 34x

69. f(x) = x3

70. f(x) = x2 + 1

71. f(x) = 1x - 1 72. f(x) = 1 - 3x73. Cost of Driving In 2010, the average cost of driving a

new car in the United States was about 50 cents per mile.Symbolically, graphically, and numerically representa function f that computes the cost in dollars of driv-ingxmiles. For the numerical representation (table) let

x= 10, 20, 30, p , 70. (Source: Associated Press.)f(x) = 0.50x;*



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74. Federal Income Taxes In 2010, the lowest U.S.income tax rate was 10 percent. Symbolically, graphi-cally, and numerically represent a functionfthat com-putes the tax on a taxable income ofxdollars. For thenumerical representation (table) let x= 1000, 2000,3000, p ,7000, and for the graphical representationlet 0 x 10,000. (Source: Internal Revenue Service.)f(x) = 0.10x;*

75. Global Web Searches The number of World WideWeb searches S in billions during year x can beapproximated by S(x) = 225x - 450,650 from 2009to 2012. Evaluate S(2011) and interpret the result.(Source: RBC Capital Markets Corp.)*

76. Cost of Smartphones The average cost difference

D in dollars between smartphones and all other

types of phones during year xcan be approximatedby D(x) = -23.5x + 47,275 from 2005 to 2009.Evaluate D(2009) and interpret the result. (Source:

Business Insider.)*

IDENTIFYING DOMAINS AND RANGES

Exercises 7784: Use the graph of f to identify its domain

and range.

Exercises 85 and 86: Use the diagram to find the domain

and range of f.

77.

x

y

-3 -2 -1 1 2 3

-3

-2

-1

1

3

78.

x

y

-3 -1 1 2 3

-3

1

2

3

79.

2 1 1 2 3 4

3

1

1

2

3

x

y 80.

x

y

-4 -2 1 2 3 4

-4

-3

1

2

3

4

81.

x

y

-2 2

-2

1

2

82.

x

y

-3

1

2

3

-3 -2 -1 1 2 3

83.

x

y

-3 -2 -1 1 2 3

-3

-2

1

2

3

84.

x

y

-6 -4 -2 2 4 6

-6

-2

2

4

6

85.1

2

3

4

5

6

7

f 86.26

32

9

f

12

13

15

Exercises 87100: Find the domain.

87. f(x) = 10xAll real numbers

88. f(x) = 5 - xAll real numbers

99. f(x) = 1x

x 0

100. f(x) = 1 - 3x2All real numbers

89. f(x) = x2 - 3All real numbers

90. f(x) = 12x2All real numbers

91. f(x) = 3x - 5

x 5

92. f(x) = xx + 1

x -1

93. f(x) = 2xx2 + 1

All real numbers

94. f(x) = 61 - x

x 1

95. f(x)

=

1x-

1x 1 96. f(x)

=

0x0 All real numbers97. f(x) = 0x - 5 0

All real numbers98. f(x) = 12 - x

x 2

101. Humpback Whales The number of humpbackwhales W sighted in Mauis annual whale census

for yearxis given by W(2005) = 649, W(2006) =1265, W(2007) = 959, W(2008) = 1726, andW(2009) = 1010. (Source: Pacific Whale Foundation.)

(a) Evaluate W(2008) and interpret the result.* 1726 (b) Identify the domain and range ofW.* (c) Describe the pattern in the data.*

D: -2 x 2 R: 0 y 2

D: -2 x 2 R: -3 y 3

D: -2 x 4 R: -2 y 2

D: All real numbers R: All real numbers

D = {1, 2, 3, 4}R = {5, 6, 7} D = {12, 13, 15}R = {9, 26, 32}

D: All real numbers R:y -1

D: -1 x 3 R: -3 y -1

D: -3 x 3 R: -3 y 2

D = { -6, -4, -2, 2, 4, 6} R = {-4, -2, 2, 4, 6}

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102. Digital Music Downloads The percentage ofdigital music D that was purchased through down-loads during year x is given by D(2004) = 0.9,D(2005) = 5.7, D(2006) = 6.7, D(2007) = 11.2,and D(2008) = 12.8. (Source: The Recording IndustryAssociation of America.)

(a) EvaluateD(2006) and interpret the result.* 6.7%

(b) Identify the domain and range ofD.*(c) Describe the pattern in the data. Increased rapidly

103. Cost of Tuition Suppose that a student can takefrom 1 to 20 credits at a college and that each creditcosts $200. If function Ccalculates the cost of taking

xcredits, determine the domain and range ofC.D = 51,2,3, p ,206 ;R = 5200,400,600, p ,40006

104. Falling Ball Suppose that a ball is dropped from awindow that is 64 feet above the ground and that theball strikes the ground after 2 seconds. If function

Hcalculates the height of the ball after tseconds,determine a domain and range forH, while the ball

is falling. D: 0 t 2;R: 0 H 64IDENTIFYING A FUNCTION

Exercises 105108: Determine whether the diagram

could represent a function.

105. No 106. Yes

1

2

4

5

6

1

2

3

8

107. Yes 108. No

a

b

c

d

e

f

5

6

10

11

109. Average Precipitation The table lists the monthlyaverage precipitation P in Las Vegas, Nevada,where x= 1 corresponds to January and x= 12corresponds to December.

Source: J. Williams.

x(month) 1 2 3 4 5 6

P (inches) 0.5 0.4 0.4 0.2 0.2 0.1

x(month) 7 8 9 10 11 12

P (inches) 0.4 0.5 0.3 0.2 0.4 0.3

Source: J. Williams.

x(month) 1 2 3 4 5 6

W(mph) 10.4 12.7 10.4 10.4 8.1 8.1

x(month) 7 8 9 10 11 12

W(mph) 6.9 6.9 6.9 8.1 9.2 9.2

Exercises 111122: Determine whether the graph repre-

sents a function. If it does, identify the domain and range.*

-4 -2 2 3 4

-4-3

-2

-1

1

2

3

4

x

y113.

-4 -2 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y

(4, 4) (4, 4)

115.

-3 -2 1 2 3

-3

-2

-1

1

2

x

y117.

(a) Determine the value ofP during May. 0.2 (b) Is P a function ofx? Explain.*

(c) IfP = 0.4, findx. 2, 3, 7, 11

110. Wind Speeds The table at the top of the next col-

umn lists the monthly average wind speed W inLouisville, Kentucky, where x= 1 corresponds toJanuary andx= 12 corresponds to December.

-4 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y

111.

-4 -2 1 2 3 4

-4

-2

1

2

3

4

x

y

(2, 4)

(2, 4)

112.

-4 -1 1 2 3 4

-4-3

-2

1

2

3

4

x

y114.

-2 1 2

-2

-1

1

2

x

y116.

-4 -2 2 4

-4

-2

4

x

y118.

(a) Determine the month with the highest averagewind speed. February

(b) Is Wa function ofx? Explain.*(c) IfW= 6.9, findx. 7, 8, 9



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x

y

-3 -2 -1 1 2 3

-2

-3

-1

1

2

3

.

x

y

-6 -4 -2 2 4 6

-4

-6

-2

2

4

6

121.

127.

x

y

128.

x

y

Exercises 123126: Determine whether S is a function.

123. S= {(1, 2), (4, 5), (7, 8), (5, 4), (2, 2)} Yes

124. S= {(4, 7), (- 2, 1), (3, 8), (4, 9)} No

125. Sis given by the table. No

126. Sis given by the table. Yes

x 5 10 5

y 2 1 0

x - 3 - 2 -1

y 10 10 10

GRAPHICAL INTERPRETATION

Exercises 127 and 128: The graph represents the distance

that a person is from home while walking on a straight

path. The x-axis represents time and the y-axis represents

distance. Interpret the graph.*

122.

129. Texting The average 18- to 24-year-old person textsabout 1500 messages per month. Sketch a graph thashows the total number of text messages sent over aperiod of 4 months. Assume that the same numberof texts is sent each day. (Source: The Nielsen Company.)*

130. Computer Viruses In 2000 there were abou

50 thousand computer viruses. In 2010 there wereabout 1.6 million computer viruses. Sketch a graphof this increase from 2000 to 2010. Answers mayvary. (Source: Symantec.)*

GRAPHING CALCULATORS

Exercises 131136: Show the given viewing rectangle

on your graphing calculator. Predict the number of tick

marks on the positive x-axis and the positive y-axis.*

131. Standard viewing rectangle 10; 10

132. [- 12, 12, 2] by [ -8, 8, 2] 6; 4

133. [0, 100, 10] by [ - 50, 50, 10] 10; 5

134. [- 30, 30, 5] by [ -20, 20, 5] 6; 4

135. [1980, 1995, 1] by [12000, 16000, 1000] 16; 5

136. [1900, 1990, 10] by [1700, 2800, 100] 10; 12

Exercises 137142: Use your calculator to make a scat

terplot of the relation after determining an appropriate

viewing rectangle.*

137. {(4, 3), (- 2, 1), ( - 3, -3), (5, - 2)}

138. {(5, 5), (2, 0), ( -2, 7), (2, - 8), (- 1, - 5)}

139. {(20, 40), ( -25, -15), ( -20, 25), (15, -25)}

140. {(-13, 12), (3, 10), ( -15, - 4), (12, -9)}

141. {(100, - 100), (50, 200), ( -150, -140), ( - 30, 80)}

142. {(-125, 75), (45, 65), (- 53, -67), (150, - 80)}

Exercises 143146: Make a table and graph of y = f(x)Let x= -3, - 2, -1, p , 3 for your table and use the stan-dard window for your graph.*

143. f(x) = 1x + 3 144. f(x) = x3 - 12x2

145. f(x) = 5 - x5 + x

146. f(x) = 0 2 - x0 + 23x

x

y

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

119.

x

y

-6 -4 -2 2 4 6

-4

-2

2

4

6

120.

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Working with Real DataGroup Activity

U.S. Craigslist Visitors The following table lists theaverage number of unique visitors to Craigslist forselected years.

(a) Make a scatterplot of the data. Letxrepresent thenumber of years after 2006. Discuss any trend innumbers of visitors to Craigslist.*

(b) Estimate the slope of a line that could be used tomodel the data. m 112,000 (answersmayvary)

(c) Find an equation of a line y = mx + b that models

the data. y = 112,000x + 180,000 (answers may vary)

(d) Interpret the slope as a rate of change. Numbers ofunique visitors increased by 112,000 per year on average.

Directions: Form a group of 2 to 4 people. Select someone to record the groups responses

for this activity. All members should work cooperatively to answer the questions. If yourinstructor asks for the results, each member of the group should be prepared to respond.

Source: Citi Investment Research and Analysis.

Year 2006 2007 2008Visitors 180,000 288,000 420,000

Year 2009 2010

Visitors 516,000 624,000

(e) Use your results to estimate the number ofuniquevisitors to Craigslist in 2012.About 852,000 (answers may vary)

WRITING ABOUT MATHEMATICS

147. Give an example of a function. Identify the domainand range of your function.

148. Explain in your own words what a function is. Howis a function different from other relations?

149. Explain how to evaluate a function by using a graph.Give an example.

150. Give one difficulty that may occur when you use atable of values to evaluate a function.

8.2 Linear Functions

Basic Concepts Representations of Linear Functions Modeling Data with Linear Functions The Midpoint Formula (Optional)

A LOOK INTO MATH N Functions are frequently used to model, or describe, the real world. For example, peopleare becoming more energy conscious. As a result, there is an increase in the number ofgreen buildings that are being constructed. Table 8.6 lists estimated U.S. sales of greenbuilding material. Because sales increase by $5 billion each year, a linear function can beused to model these data. (See Example 7.) In this section we discuss this important typeof function.

TABLE 8.6 Green Material Sales ($ billions)

Source: Freedonia Group, Green Building Material.

Year 2010 2011 2012 2013

Sales 65 70 75 80

NEW VOCABULARY

n Linear functionn Rate of changen Constant functionn Midpoint



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5238.2 LINEAR FUNCTIONS

Another graph ofy = f(x) with a differenty-scale is shown in Figure 8.29 on the nexpage. Because they-values always decrease by the same amount for each 1-hour increaseon thex-axis, we say that function f has a constant rate of change. In this example, the constant rate of change is - 2F per hour.

Basic Concepts

TABLE 8.7 House Temperature

Elapsed Time (hours) 0 1 2 3 4 5

Temperature (F) 80 78 76 74 72 70

1-hour increaseQ

2F decreaseQ

We want to determine a function f that models, or calculates, the house temperatureafterxhours. To do this, we will find numerical, graphical, verbal, and symbolic representa-tions off.

NUMERICAL REPRESENTATION (TABLE OF VALUES) We can think of Table 8.7 asa numerical representation (table of values) for the function f. A similar numerical repre-sentation that usesxand f(x) is shown in Table 8.8.

STUDY TIP

Be sure you understand whatrepresentations of a functionare.

GRAPHICAL REPRESENTATION (GRAPH) To graph y = f(x), we begin by plot-ting the points in Table 8.8, as shown in Figure 8.27. This scatterplot suggests that a linemodels these data, as shown in Figure 8.28. We call f a linear function because its graphis a line.

Figure 8.27 A Scatterplot

10 2 3 4 5 6

60

70

80

90

100

x

y

Elapsed Time (hours)

Temperature(F)

House Temperature

Figure 8.28 A Linear Function

10 2 3 4 5 6

60

70

80

90

100

x

y


Te

mperature(F)

y =f(x)

Graphical Representation off(x)

N REAL-WORLD CONNECTION Suppose that the air conditioner is turned on when the temperature inside a house is 80F. The resulting temperatures are listed in Table 8.7 for variouselapsed times. Note that for each 1-hour increase in elapsed time, the temperature decreasesby 2F.

TABLE 8.8 Numerical Representation off(x)

x 0 1 2 3 4 5

f(x) 80 78 76 74 72 70

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VERBAL REPRESENTATION (WORDS) Over a 5-hour period, the air conditioner low-ers the initial temperature of 80F by 2F for each elapsed hourx. Thus a description of howto calculate the temperature is:

Multiplyxby - 2F and then add 80F. Verbal representation off(x)

SYMBOLIC REPRESENTATION(FORMULA) Our verbal representation off(x) makesit straightforward for us to write a formula.

Elapsed Hours

f(x) = 2 x + 80 Symbolic representation off(x)Rate of Change Initial Temperature

For example,

f(2.5) = - 2(2.5) + 80 = 75means that the temperature is 75F after the air conditioner has run for 2.5 hours. In thisinstance, it might be appropriate to limit the domain off to x-values between 0 and 5,inclusive.

LINEAR FUNCTION

A function fdefined by f(x) = mx + b, where m and b are constants, is a linearfunction.

For f(x) = 2x + 80, we have m = 2 and b = 80. The constant m represents therate at which the air conditioner cools the building, and the constant b represents the initialtemperature.

NOTE: The value ofm represents the slope of the graph off(x) = mx + b, and b is they-intercept.

N REAL-WORLD CONNECTION In general, a linear function defined by f(x) = mx + b changesby m units for each unit increase in x. This rate of change is an increase if m 7 0 and adecrease if m 6 0. For example, if new carpet costs $20 per square yard, then the linearfunction defined by C(x) = 20x gives the cost of buying x square yards of carpet. Thevalue ofm = 20 gives the cost (rate of change) for each additional square yard of carpet.For function C, the value ofb is 0 because it costs $0 to buy 0 square yards of carpet.

Figure 8.29

10 2 3 4 5 6

70

72

74

76

78

80

x

y


T

emperature(F)

y =f(x)2

1

2

1

y-values decrease by 2

for each 1-hour increase

in thex-values.

Constant Rate of Change



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NOTE: Iffis a linear function, then f(0) = m(0) + b = b. Thus b can be found by evaluating f(x) at x= 0.

EXAMPLE 1 Identifying linear functions

Determine whetherfis a linear function. Iffis a linear function, find values for m and b sothat f(x) = mx + b.(a) f(x) = 4 - 3x (b) f(x) = 8 (c) f(x) = 2x2 + 8Solution(a) Let m = - 3 and b = 4. Then f(x) = -3x + 4, andfis a linear function.(b) Let m = 0 and b = 8. Then f(x) = 0x + 8, andfis a linear function.(c) Functionfis not linear because its formula containsx2. The formula for a linear func-

tion cannot contain anxwith an exponent other than 1.


EXAMPLE 2 Determining linear functions

Use each table of values to determine whether f(x) could represent a linear function. Iffcould be linear, write a formula forfin the form f(x) = mx + b.(a) (b)

(c) (d)

Solution(a) For each unit increase in x, f(x) increases by 5 units, so f(x) could be linear with

m = 5. Because f(0) = 10, b = 10. Thus f(x) = 5x + 10.(b) For each 2-unit increase inx, f(x) decreases by 2 units. Equivalently, each unit increase

inxresults in a 1-unit decrease in f(x), so f(x) could be linear with m = -1. Becausef(0) = 2, b = 2. Thus f(x) = -x + 2.

(c) Each unit increase inxdoes not result in a constant change in f(x). Thus f(x) does norepresent a linear function.

(d) For any change inx,f(x) does notchange, so f(x) could be linear with m = 0. Becausef(0) = 7, let b = 7. Thus f(x) = 0x + 7, or f(x) = 7. (When m = 0, we say that fis a constant function. See Example 8.)

Now Try Exercises 23, 25, 27, 31

Representations of Linear FunctionsThe graph of a linear function is a line. To graph a linear functionfwe can start by makinga table of values and then plotting three or more points. We can then sketch the graph offby drawing a line through these points, as demonstrated in the next example.

x 0 1 2 3

f(x) 10 15 20 25 x - 2 0 2 4

f(x) 4 2 0 -2

x 0 1 2 3

f(x) 1 2 4 7 x - 2 0 3 5

f(x) 7 7 7 7

READING CHECK

Explain what a linear function is and what its graph looks like.

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EXAMPLE 3 Graphing a linear function by hand

Sketch a graph off(x) = x - 1. Use the graph to evaluate f( -2).SolutionBegin by making a table of values containing at least three points. Pick convenient values of

x, such asx= 1,0,1.

f( 1) = 1 - 1 = 2f(0) = 0 - 1 = 1f(1) = 1 - 1 = 0

Display the results, as shown in Table 8.9.Plot the points ( 1, 2), (0, 1), and (1, 0). Sketch a line through these points to

obtain the graph off. A graph of a line results when infinitely many points are plotted, asshown in Figure 8.30.

To evaluate f(- 2), first find x= 2 on the x-axis. See Figure 8.31. Then movedownward to the graph off. By moving across to they-axis, we see that the corresponding

y-value is 3. Thus f( 2) = 3.

Now Try Exercises 39, 57

TABLE 8.9

x y

1 2

0 1

1 0

Figure 8.30

3 3

3

1

3

x

y

(1, 0)

(0, 1)(1, 2)

y =x 1

Figure 8.31

2

3

1

3

x

y

y =x 1

f(2) = 3

In the next example a graphing calculator is used to create a graph and table.

EXAMPLE 4 Using a graphing calculator

Give numerical and graphical representations off(x) = 12x - 2.SolutionNumerical Representation To make a numerical representation, construct the table forY1 = .5X - 2, starting at x= -3 and incrementing by 1, as shown in Figure 8.32(a).(Other tables are possible.)Graphical Representation Graph Y1 in the standard viewing rectangle, as shown in

Figure 8.32(b). (Other viewing rectangles may be used.)

Now Try Exercise 75

Figure 8.32

(a)

Y1.5X2

X Y13 3.52 31 2.50 21 1.52 13 .5

(b)

y = x 21

2

[-10, 10, 1] by [ - 10, 10, 1]

CALCULATOR HELP

To make a table, see Appendix A

(pages AP-2 and AP-3). To

make a graph, see Appendix A

(page AP-5).



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TABLE 8.10

x f(x)

-1 5

0 2

1 - 1

Figure 8.33

4 2

2

1

6

x

y

(1, 1)

(0, 2)

(1, 5) y = 3x + 2

EXAMPLE 5 Representing a linear function

A linear function is given by f(x) = -3x + 2.(a) Give a verbal representation off.(b) Make a numerical representation (table) offby lettingx= -1, 0, 1.(c) Plot the points listed in the table from part (b). Then sketch a graph ofy = f(x).Solution(a) Verbal Representation Multiply the inputxby -3 and add 2 to obtain the output.

(b) Numerical Representation Evaluate the formula f(x) = -3x + 2 at x= - 1, 0, 1which results in Table 8.10. Note that f( -1) = 5, f(0) = 2, and f(1) = - 1.

(c) Graphical Representation To make a graph offby hand without a graphing calculator

plot the points ( -1, 5), (0, 2), and (1, - 1) from Table 8.10. Then draw a line passingthrough these points, as shown in Figure 8.33.

Now Try Exercise 71

NOTE: To graph y = -3x + 2 in Example 5, we could also graph a line with slope 3

andy-intercept 2.

MAKING CONNECTIONS

Mathematics in NewspapersThink of the mathematics that you seein newspapers or in online publications.Often, percentages are described ver-bally, numbers are displayed in tables,and data are shown in graphs. Seldomareformulas given, which is an impor-

tant reason to study verbal, numerical,and graphical representations.

CRITICAL THINKING

Two points determine a line. Why is it a good idea to plot at least three points when graphing linear function by hand?

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Modeling Data with Linear Functions

MODELING DATA WITH A LINEAR FUNCTION

The formula f(x) = mx + b may be interpreted as follows. f(x) = mx + b

(New amount) = (Change) + (Fixed amount)

Whenxrepresents time, change equals (rate of change) * (time).

f(x) = m * x + b(Future amount) = (Rate of change) * (Time) + (Initial amount)

EXAMPLE 6 Modeling growth of bamboo

Bamboo is gaining popularity as a green building material because of its fast-growing, regen-erative characteristics. Under ideal conditions, some species of bamboo grow at an astonish-ing 2 inches per hour. Suppose a bamboo plant is initially 6 inches tall. (Source:Cali Bamboo.)(a) Find a functionHthat models the plants height in inches under ideal conditions after thours.(b) FindH(3) and interpret the result.

Solution(a) The initial height is 6 inches and the rate of change is 2 inches per hour.

H(t) = 2 * t + 6,

(Futureheight) = (Rateofchange) * (Time) + (Initialheight)orH(t) = 2t + 6.

(b)H(3) = 2(3) + 6 = 12. After 3 hours the bamboo plant is 12 inches tall.


N REAL-WORLD CONNECTION A distinguishing feature of a linear function is that when theinputxincreases by 1 unit, the output f(x) = mx + b always changes by an amount equalto m. For example, the percentage of wireless households during yearxfrom 2005 to 2010can be modeled by the linear function

f(x) = 4x - 8013,wherexis the year. The value of m = 4 indicates that the percentage of wireless house-holds has increased, on average, by 4% per year. (Source: National Center for Health Statistics.)

The following are other examples of quantities that are modeled by linear functions.Try to determine the value of the constant m.

The wages earned by an individual workingxhours at $8 per hour m = 8 The distance traveled by a jet airliner inxhours if its speed is 500 miles per hour m = 500 The cost of tuition and fees when registering for xcredits if each credit costs $200 and

the fees are fixed at $300 m = 200

When we are modeling data with a linear function defined by f(x) = mx + b, the fol-lowing concepts are helpful to determine m and b.

N REAL-WORLD CONNECTION These concepts are applied in the next three examples.



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READING CHECK

How can you determinewhether data in a table canbe modeled by a linearfunction?

(a) Make a scatterplot of the data and sketch the graph of a function f that models thesedata. Letxrepresent years after 2010. That is, let x= 0 correspond to 2010, x= 1 to

2011, and so on.(b) What were the sales in 2010? What was the annual increase in sales each year?(c) Find a formula for f(x).(d) Use your formula to estimate sales in 2014.

Solution(a) In Figure 8.34 the scatterplot suggests that a linear function models the data. A line hasbeen sketched with the data.

EXAMPLE 7 Modeling demand for building green

Table 8.11 lists estimated sales of green building material in billions of dollars. (Refer to ALook Into Math at the beginning of this section.)

Figure 8.34 A Linear Model

10 2 3 4

60

70

80

x

y

Year (0 2010)

Sales($billions)

y =f(x)

Green Building Material Sales

TABLE 8.11 Green Material Sales ($ billions)

Source: Freedonia Group, Green Building Material.

Year 2010 2011 2012 2013

Sales 65 70 75 80

(b) From Table 8.11, sales for green material were $65billion in 2010, with sales increasing at a constant rate of $5 billion per year.

(c) From part (b) initial sales (x= 0) were $65 billion, and sales increased by $5 billionper year. Thus

f(x) = 5 * x + 65,(Futuresales) = (Rateofchangeinsales) * (Time) + (Initialsales)

or f(x) = 5x + 65.(d) Becausex= 4 corresponds to 2014, evaluate f(4).

f(4) = 5(4) + 65 = 85This model estimates sales of green building material to be $85 billion in 2014.


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In the next example, we consider a simple function that models the speed of a car.

EXAMPLE 8 Modeling with a constant function

A car travels on a freeway with its speed recorded at regular intervals, as listed in Table 8.12.

(a) Discuss the speed of the car during this time interval.(b) Find a formula for a functionfthat models these data.(c) Sketch a graph offtogether with the data.

Solution(a) The speed of the car appears to be constant at 70 miles per hour.(b) Because the speed is constant, the rate of change is 0. Thus

f(x) = 0x + 70(Future speed) = (Change in speed) + (Initial speed)

and f(x) = 70. We callfa constant function.(c) Becausey = f(x), graphy = 70 with the data points

(0, 70), (1, 70), (2, 70), (3, 70), and (4, 70)

to obtain Figure 8.35.

TABLE 8.12 Speed of a Car

Elapsed Time (hours) 0 1 2 3 4

Speed (miles per hour) 70 70 70 70 70


CRITICAL THINKING

Find a formula for a func-tion Dthat calculates thedistance traveled by the carin Example 8 afterxhours.What is the rate of changeforD(x)?

Figure 8.35 Speed of a Car

10 2 3 4 5

10

20

30

40

50

60

70

80

x

y

Time (hours)

Speed(mph)

y = 70

Constant Function

The function defined by f(x) = 70 is an example of a constant function. Aconstant function is a linear function with m

=0 and can be written as f

(x)

=b.Regardless of the input, a constant function always outputs the same value, b. Its graph

is a horizontal line. Its domain is all real numbers and its range is R = {b}.

N REAL-WORLD CONNECTION The following are three applications of constant functions.

A thermostat calculates a constant function regardless of the weather outside by main-taining a set temperature.

A cruise control in a car calculates a constant function by maintaining a fixed speed,regardless of the type of road or terrain.

A constant function calculates the 1250-foot height of the Empire State Building.



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The Midpoint Formula (Optional)N REAL-WORLD CONNECTION A common way to make estimations is to average data items

For example, in 2000 the average tuition and fees at public two-year colleges were about$1700, and in 2010 they were about $2700. (Source: The College Board.) To estimate tuition andfees in 2005, we could average the 2000 and 2010 amounts.

1700 + 2700

2= +

2200 Finding the average

This technique predicts that tuition and fees were $2200 in 2005 and is referred to as finding the midpoint.

MIDPOINT FORMULA ON THE REAL NUMBER LINE The midpoint of a line segment is the unique point on the line segment that is an equal distance from the endpoints

For example, in Figure 8.36 the midpointMof 3 and 5 on the real number line is 1.

Figure 8.36

4 3 2 1 10 2 3 4 5 6

M

We can calculate the value ofMas follows.

M=x1 + x2

2=

3 + 5

2= 1

MIDPOINT FORMULA IN THE xy-PLANE The midpoint of a line segment in thexy-plane can be found in a similar way. Figure 8.37(a) shows the midpoint on the line

segment connecting the points (x1,y1) and (x2,y2). Thex-coordinate ofMis equal to theaverage ofx1 and x2, and they-coordinate ofMis equal to the average ofy1 and y2. Forexample, the line segment with endpoints ( 2, 1) and (4, 3) is shown in Figure 8.37(b)The coordinates of the midpoint are

M= a2 + 4

2 ,

1 + ( 3)

2 b = (1, 1).

Average thex-values

to find the midpoint.

Q

This discussion is summarized as follows.

MIDPOINT FORMULA IN THE xy-PLANE

The midpoint of the line segment with endpoints (x1,y1) and (x2,y2) in thexy-plane is

ax1 + x22

,y1 + y2

2b .

x

y

x1 x2x1 +x2

2

y1

y2

y1 +y2

2

(x2,y2)

M

(x1,y1) 2 1 1 2 3 4

3

2

1

1

2

3

x

y

(2, 1)

(4, 3)

M= (1, 1)

(a) (b)

Figure 8.37

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EXAMPLE 9 Finding the midpoint

Find the midpoint of the line segment connecting the points (3, 2) and (4, 1).

SolutionIn the midpoint formula let ( 3, 2) be (x1,y1) and (4, 1) be (x2,y2).

M= ax1 + x22 ,y1 + y2

2 b Midpoint formula

= a 3 + 42

,2 + 1

2b Substitute.

= a 12

,1

2b Simplify.

The midpoint of the line segment is 112, -122.

Now Try Exercise 93

N REAL-WORLD CONNECTION In the next example we use the midpoint formula to estimatethe divorce rate in the United States in 2005.

EXAMPLE 10 Estimating the U.S. divorce rate

The divorce rate per 1000 people in 2000 was 4.2, and in 2010 it was 3.4. (Source: StatisticalAbstract of the United States.)

(a) Use the midpoint formula to estimate the divorce rate in 2005.(b) Could the midpoint formula be used to estimate the divorce rate in 2003? Explain.

Solution(a) In the midpoint formula, let (2000, 4.2) be (x1,y1) and let (2010, 3.4) be (x2,y2).

M= ax1 + x22

,y1 + y2

2b Midpoint formula

= a 2000 + 20102

,4.2 + 3.4

2b Substitute.

= (2005, 3.8) Simplify.

The midpoint formula estimates that the divorce rate was 3.8 per 1000 people in 2005.(Note that the actual rate was 3.6.)

(b) No, the midpoint formula can only be used to estimate data that are exactly halfwaybetween two given data points. Because the year 2003 is not exactly halfway between2000 and 2010, the midpoint formula cannot be used.


NOTE: An estimate obtained from the midpoint formula is equal to an estimate obtainedfrom a linear function whose graph passes through the endpoints of the line segment. Thisfact is illustrated in the next example.



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EXAMPLE 11 Relating midpoints to linear functions

The graph of a linear function fshown in Figure 8.38 passes through the points (1, 3

and (2, 3).

(a) Find a formula forf(x).(b) Evaluate f1122. Does your answer agree with the graph?(c) Find the midpoint Mof the line segment connecting the points (1, 3) and (2, 3)

Comment on your result.

Solution(a) The graph offis a line that passes through (1, 3) and (2,3). The slope m of the line is

m =- 3 - 3

2 - (- 1)= -

6

3= 2,

and from the graph, they-intercept is 1. Thus f(x) = 2x + 1.(b) f1122 = - 21122 + 1 = 0. Yes, they agree because the point 112, 02 lies on the graph

ofy = f(x) in Figure 8.38.(c) The midpoint of the line segment connecting ( 1, 3) and (2, 3) is

M= a 1 + 22

,3 + ( 3)

2b = a 1

2, 0b .

Finding the midpoint M= 112, 02 of the line segment with endpoints (1, 3) and(2, 3) is equivalent to evaluating the linear function f, whose graph passes through(1, 3) and (2, 3), atx= 12.


x

y

(2, 3)

(1, 3)yf(x)

4 2 2 4

1

4

4

2

M

Figure 8.38

Linear Function Can be represented by f(x) = mx + bIts graph is a line with slope m and

y-intercept b.

f(x) = 2x - 6, m = 2 and b = -6f(x) = 10, m = 0 and b = 10


Constant Function Can be represented by f(x) = bIts graph is a horizontal line.

f(x) = -7, b = - 7f(x) = 22, b = 22

Rate of Change for aLinear Function

The output of a linear functionchanges by a constant amount foreach unit increase in the input.

f(x) = - 3x + 8 decreases 3 units foreach unit increase inx.f(x) = 5 neither increases nor decreases.The rate of change is 0.

8.2 Putting It All Together

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Symbolic Mathematical formula in the form

f(x) = mx + bf(x) = 2x + 1,where m = 2 and b = 1

REPRESENTATION COMMENTS EXAMPLE

Verbal Multiply the inputxby m and add b. Multiply the inputxby 2 and then add 1to obtain the output.

Numerical (table ofvalues)

For each unit increase inxin thetable, the output off(x) = mx + bchanges by an amount equal to m.

Graphical The graph of a linear function is aline. Plot at least 3 points and thensketch the line.

Iff(x) = mx + b, then the graph offhas slope m andy-intercept b.

x

y

y = 2x + 1

2 1 2 3 4

1

2

3

4

5

1

1-unit increase

2-unit increase

Q Q

Q

x 0 1 2

f(x) 1 3 5


Midpoint Formula The midpoint of the line segment

connecting (x1,y1) and (x2,y2) is

ax1 + x22

,y1 + y2

2b .

The midpoint of the line segment

connecting (2, 3) and (4, 5) is

a -2 + 42

,3 + 5

2b = (1, 4).


continued from previous page

8.2 Exercises

CONCEPTS AND VOCABULARY

1. The formula for a linear function is f(x) = .mx + b

2. The formula for a constant function is f(x) = b .3. The graph of a linear function is a(n) line .

4. The graph of a constant function is a(n) line. horizontal5. If f(x) = 7x + 5, each time x increases by 1 unit,

f(x) increases by 7 units.

6. If f(x) = 5, each time x increases by 1 unit, f(x)increases by 0 units.

7. (True or False?) Every constant function is a linear

function. T

8. (True or False?) Every linear function is a constantfunction. F

Q



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Exercises 2332: (Refer to Example 2.) Use the table to

determine whether f(x) could represent a linear function. If it could, write f(x) in the form f(x) = mx + b.23. Yes; f(x) = 3x - 6

24. Yes; f(x) = 2x - 2

25. Yes; f(x) = - 32x + 3

26. No

27. No

28. Yes; f(x) = -3x

29. Yes; f(x) = 2x

30. No

31. Yes; f(x) = -4

32. Yes; f(x) = 5

EVALUATING LINEAR FUNCTIONS

Exercises 3338: Evaluate f(x) at the given values of x.33. f(x) = 4x x= -4, 5 -16; 2034. f(x) = - 2x + 1 x= -2, 3 5; -535. f(x) = 5 - x x= - 23, 3 173 ; 236. f(x) = 12x - 14 x= 0, 12 - 14; 0

9. IfC(x) = 2xcalculates the cost in dollars of buyingxsquare feet of carpet, what does 2 represent in theformula? Interpret the fact that the point (10, 20) lieson the graph ofC.*

10. If G(x) = 100 - 4x calculates the number of gal-lons of water in a tank afterxminutes, what does -4

represent in the formula? Interpret the fact that thepoint (5, 80) lies on the graph ofG.*

IDENTIFYING LINEAR FUNCTIONS

Exercises 1118: Determine whether f is a linear func-

tion. If f is linear, give values for m and b so that f may be

expressed as f(x) = mx + b.11. f(x) = 1

2x - 6 Yes; m = 12, b = -6

12. f(x) = x Yes; m = 1, b = 0

13. f(x) = 52

- x2 No

14. f(x) = 1x + 3 No15. f(x) = -9 Yes; m = 0, b = -916. f(x) = 1.5 - 7.3x Yes; m = -7.3, b = 1.517. f(x) = -9x Yes; m = -9, b = 0

18. f(x) = 1x

No

Exercises 1922: Determine whether the graph repre-

sents a linear function.

19. Yes 20. No

-4 -2 2 4

-4

-2

2

4

x

y

-4 -2 2 4

-4

2

4

x

y

21. No 22. Yes

-4 -2 2 4

-4

-2

2

4

x

y

-4 -2 2 4

-4

2

4

x

y

x 0 1 2 3

f(x) - 6 -3 0 3

x 0 2 4 6

f(x) - 2 2 6 10

x - 2 0 2 4

f(x) 6 3 0 - 3

x 0 3 6 9

f(x) 8 4 2 1

x - 2 -1 0 1

f(x) 6 3 0 - 3

x -2 -1 0 1

f(x) -5 0 20 40

x -1 0 1 2

f(x) -4 - 4 - 4 -4

x 1 2 3 4

f(x) 0 1 3 7

x 0 2 3 4

f(x) 0 4 6 8

x 2 5 6 8

f(x) 5 5 5 5

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37. f(x) = -22 x= - 34, 13 -22; -2238. f(x) = 9x - 7 x= -1.2, 2.8 -17.8; 18.2

Exercises 3944: Use the graph of f to evaluate the given

expressions.

39. f( -1) and f(0) -2; 0 40. f( -2) and f(2) 4; 0

x

y

-2

1

2

-2 -1 1 2

x

y

-4

-3

-2

-1

1

2

4

-4 -2 1 2 4

41. f( -2) and f(4) -1; -4 42. f(0) and f(3) -1; 1

x

y

-4 -1 1 2 3 4

-4

-3

1

2

3

4

x

y

-3 -2 -1 2 3

-3

-2

1

2

3

43. f( -3) and f(1) 44. f(1.5) and f(0.5p)1; 1 -2; -2

2 1 1 2

2

1

2

x

y

2 1 1 2

1

1

2

x

y

Exercises 4548: Use the verbal description to write a

formula for f(x). Then evaluate f(3).45. Multiply the input by 6. f(x) = 6 x; 18

46. Multiply the input by -3 and add 7.f(x) = -3x + 7; -2

47. Divide the input by 6 and subtract 12. f(x) = x6 - 12; 0

48. Output 8.7 for every input. f(x) = 8.7; 8.7

REPRESENTING LINEAR FUNCTIONS

Exercises 4952: Match f(x) with its graph (a.d.) at thetop of the next column.

49. f(x) = 3x d. 50. f(x) = - 2x c.

51. f(x) = x - 2 b. 52. f(x) = 2x + 1 a. a. b.

-4 -2

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Beginning and Intermediate Algebra_3e_Ch08(1)