2.6 rational functions - ths advanced...

184 Chapter 2 Polynomial and Rational Functions

What you should learn• Find the domains of rational

functions.

• Find the horizontal and verticalasymptotes of graphs of rational functions.

• Analyze and sketch graphs ofrational functions.

• Sketch graphs of rational functions that have slantasymptotes.

• Use rational functions to modeland solve real-life problems.

Why you should learn itRational functions can be usedto model and solve real-lifeproblems relating to business.For instance, in Exercise 79 onpage 196, a rational function isused to model average speedover a distance.

Rational Functions2.6

IntroductionA rational function can be written in the form

where and are polynomials and is not the zero polynomial.In general, the domain of a rational function of includes all real numbers

except -values that make the denominator zero. Much of the discussion ofrational functions will focus on their graphical behavior near the -values excludedfrom the domain.

Finding the Domain of a Rational Function

Find the domain of and discuss the behavior of near any excluded

-values.

SolutionBecause the denominator is zero when the domain of is all real numbersexcept To determine the behavior of near this excluded value, evaluate

to the left and right of as indicated in the following tables.

Note that as approaches 0 from the left, decreases without bound. Incontrast, as approaches 0 from the right, increases without bound. Thegraph of is shown in Figure 2.36.

FIGURE 2.36

Now try Exercise 1.

x−1 1 2

1

2

−1

f x( ) = 1x

y

ff �x�xf �x�x

x � 0,f �x�fx � 0.

fx � 0,

x

ff �x� �1x

xx

xD�x�D�x�N�x�

f �x� �N(x)

D(x)

Note that the rational functiongiven by is alsoreferred to as the reciprocal function discussed in Section 1.6.

f �x� � 1�x

Example 1

x 0

��1000�100�10�2�1f �x�

�0.001�0.01�0.1�0.5�1

x 0 0.001 0.01 0.1 0.5 1

1000 100 10 2 1�f �x�

Mike Powell /Getty Images

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Horizontal and Vertical AsymptotesIn Example 1, the behavior of near is denoted as follows.

as as

decreases without bound increases without boundas approaches 0 from the left. as approaches 0 from the right.

The line is a vertical asymptote of the graph of as shown in Figure 2.37.From this figure, you can see that the graph of also has a horizontal asymptote—the line This means that the values of approach zero as increases or decreases without bound.

0 as 0 as

approaches 0 as approaches 0 as decreases without bound. increases without bound.

Eventually (as or ), the distance between the horizon-tal asymptote and the points on the graph must approach zero. Figure 2.38 showsthe horizontal and vertical asymptotes of the graphs of three rational functions.

The graphs of in Figure 2.37 and inFigure 2.38(a) are hyperbolas. You will study hyperbolas in Section 10.4.

f �x� � �2x � 1��x � 1�f �x� � 1�x

��x�x

xf �x�xf �x�

�xf �x��xf �x�

xf �x� � 1�xy � 0.f

f,x � 0

xxf �x�f �x�

0�x�f �x�0�x��f �x�

x � 0f

Section 2.6 Rational Functions 185

Additional Examples

State the domain of each function.

a.

b.

Solution

a. The domain is all real numbersexcept

b. The domain is all real numbersexcept and x � 6.x � �2

x � �10.

f �x� �x � 1

�x � 2��x � 6�

f �x� �3x

x � 10

x

f(x) = 2x + 1x + 1

Horizontalasymptote:

y = 2

Verticalasymptote:

x = −1

−2−3 1

1

3

4

y

−1

2

(a)

FIGURE 2.38

x−2 −1 1 2

1

3

2

f(x) = 4x + 12


y = 0

y

(b)

x32−1

2

3

4

f (x) =(x −1)2

Verticalasymptote:

x = 1Horizontalasymptote:

y = 0

y2

1

(c)

x

f(x) = 1x

−1−2 1 2

1

2

−1 Horizontalasymptote:

y = 0

Verticalasymptote:

x = 0

y

FIGURE 2.37

Definitions of Vertical and Horizontal Asymptotes

1. The line is a vertical asymptote of the graph of if

or

as either from the right or from the left.

2. The line is a horizontal asymptote of the graph of if

as or ��.x�x

bf �x�

fy � b

a,x

��f �x��f �x�

fx � a

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Finding Horizontal and Vertical Asymptotes

Find all horizontal and vertical asymptotes of the graph of each rational function.

a. b.

Solution

a. For this rational function, the degree of the numerator is equal to the degreeof the denominator. The leading coefficient of the numerator is 2 and the lead-ing coefficient of the denominator is 1, so the graph has the line as ahorizontal asymptote. To find any vertical asymptotes, set the denominatorequal to zero and solve the resulting equation for

Set denominator equal to zero.

Factor.

Set 1st factor equal to 0.

Set 2nd factor equal to 0.

This equation has two real solutions and so the graph has thelines and as vertical asymptotes. The graph of the function isshown in Figure 2.39.

b. For this rational function, the degree of the numerator is equal to the degree of the denominator. The leading coefficient of both the numerator and denom-inator is 1, so the graph has the line as a horizontal asymptote. To findany vertical asymptotes, first factor the numerator and denominator as follows.

By setting the denominator (of the simplified function) equal to zero,you can determine that the graph has the line as a vertical asymptote.

Now try Exercise 9.

x � 3x � 3

f �x� �x2 � x � 2x2 � x � 6

��x � 1��x � 2��x � 2��x � 3� �

x � 1x � 3

, x � 2

y � 1

x � 1x � �1x � 1,x � �1

x � 1 x � 1 � 0

x � �1 x � 1 � 0

�x � 1��x � 1� � 0

x2 � 1 � 0

x.

y � 2

f �x� �x2 � x � 2x2 � x � 6

f �x� �2x2

x2 � 1


Asymptotes of a Rational FunctionLet be the rational function given by

where and have no common factors.

1. The graph of has vertical asymptotes at the zeros of

2. The graph of has one or no horizontal asymptote determined bycomparing the degrees of and

a. If the graph of has the line (the -axis) as a horizontalasymptote.

b. If the graph of has the line (ratio of the leadingcoefficients) as a horizontal asymptote.

c. If the graph of has no horizontal asymptote.fn > m,

y � an�bmfn � m,

xy � 0fn < m,

D�x�.N�x�f

D�x�.f

D�x�N�x�

f �x� �N�x�D�x�

�anxn � an�1x

n�1 � . . . � a1x � a0

bmxm � bm�1xm�1 � . . . � b1x � b0

f

Verticalasymptote:

x = −1

Horizontalasymptote: y = 2

Verticalasymptote:

x = 1

x

y

4

3

2

1

−2−3−4 2 3 4

2x2

x2 − 1f (x) =

1−1

FIGURE 2.39

Example 2

Additional Example

Identify all horizontal and verticalasymptotes of the graph of

Solution

The function has a horizontal asymptoteat because the degree of thenumerator is less than the degree of thedenominator. The function has twovertical asymptotes at and

the zeros of the denominator.x � �3,x � 3

y � 0

f �x� �x

x2 � 9.

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Analyzing Graphs of Rational FunctionsTo sketch the graph of a rational function, use the following guidelines.

Guidelines for Analyzing Graphs of Rational FunctionsLet where and are polynomials.

1. Simplify if possible.

2. Find and plot the -intercept (if any) by evaluating

3. Find the zeros of the numerator (if any) by solving the equationThen plot the corresponding -intercepts.

4. Find the zeros of the denominator (if any) by solving the equationThen sketch the corresponding vertical asymptotes.

5. Find and sketch the horizontal asymptote (if any) by using the rule forfinding the horizontal asymptote of a rational function.

6. Plot at least one point between and one point beyond each -intercept andvertical asymptote.

7. Use smooth curves to complete the graph between and beyond thevertical asymptotes.

x

D�x� � 0.

xN�x� � 0.

f �0�.y

f,

D�x�N�x�f �x� � N�x��D�x�,

Some graphing utilities have difficulty graphing rational functions that have vertical asymptotes. Often, the utility will connect parts of the graph that are not supposed to be connected. For instance, the topscreen on the right shows the graph of

Notice that the graph should consist of two unconnected portions—one to the left of and the other to the right of To eliminate this problem, you can try changing the mode of the graphing utility to dot mode. The problem with this is that the graph is then represented as a collection of dots (as shown in the bottom screen on the right) rather thanas a smooth curve.

−5

−5

5

5x � 2.x � 2

f �x� �1

x � 2.

−5

−5

5

5

Techno logy

You may also want to test for symmetry when graphingrational functions, especially forsimple rational functions. Recallfrom Section 1.6 that the graph of

is symmetric with respect to theorigin.

f �x� �1

x

The concept of test intervals from Section 2.2 can be extended to graphingof rational functions. To do this, use the fact that a rational function can changesigns only at its zeros and its undefined values (the -values for which itsdenominator is zero). Between two consecutive zeros of the numerator andthe denominator, a rational function must be entirely positive or entirely negative.This means that when the zeros of the numerator and the denominator of arational function are put in order, they divide the real number line into testintervals in which the function has no sign changes. A representative -value ischosen to determine if the value of the rational function is positive (the graph liesabove the -axis) or negative (the graph lies below the -axis).xx

x

x

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Sketching the Graph of a Rational Function

Sketch the graph of and state its domain.

Solutiony-intercept: because

x-intercept: None, because

Vertical asymptote: zero of denominator

Horizontal asymptote: because degree of degree of

Additional points:

By plotting the intercepts, asymptotes, and a few additional points, you can obtainthe graph shown in Figure 2.40. The domain of is all real numbers except

Now try Exercise 27.


Sketch the graph of

and state its domain.

Solutiony-intercept: None, because is not in the domain

x-intercept: because



Additional points:

By plotting the intercepts, asymptotes, and a few additional points, you can obtainthe graph shown in Figure 2.41. The domain of is all real numbers except


x � 0.xf

D�x�N�x� �y � 2,

x � 0,

2x � 1 � 0�12, 0�,

x � 0

f �x� �2x � 1

x

x � 2.xg

D�x�N�x� <y � 0,

x � 2,

3 � 0

g�0� � �32�0, �3

2�,

g�x� �3

x � 2You can use transformations to help you sketch graphs ofrational functions. For instance,the graph of in Example 3 is avertical stretch and a right shiftof the graph of because

� 3f �x � 2�.

� 3� 1x � 2�

g�x� �3

x � 2

f �x� � 1�x

g

4 6

2

4

−2

−4

x

Verticalasymptote:

x = 2


y = 0

g(x) =x − 2

3y

2

FIGURE 2.40

3

3 4

1

1 2−1

−1

−2

−4 −3 −2x

f x( ) = x2 1x −


= 2y

Verticalasymptote:

= 0x

y

2

FIGURE 2.41

Test Representative Value of g Sign Point oninterval x-value graph

Negative

3 Positive �3, 3�g�3� � 3�2, ��

��4, �0.5�g��4� � �0.5�4��, 2�

Test Representative Value of f Sign Point oninterval x-value graph

Positive

Negative

4 Positive �4, 1.75�f �4� � 1.75�12, ��

�14, �2�f �1

4� � �214�0, 12�

��1, 3�f ��1� � 3�1��, 0�

Example 3

Example 4

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Sketch the graph of

Solution

Factoring the denominator, you have

y-intercept: because

x-intercept:

Vertical asymptotes: zeros of denominator


Additional points:

The graph is shown in Figure 2.42.


A Rational Function with Common Factors

Sketch the graph of

SolutionBy factoring the numerator and denominator, you have


x-intercept: because

Vertical asymptote: zero of (simplified) denominator


Additional points:

The graph is shown in Figure 2.43. Notice that there is a hole in the graph atbecause the function is not defined when


x � 3.x � 3

D�x�N�x� �y � 1,

x � �1,

f ��3� � 0��3, 0�,f �0� � 3�0, 3�,

f �x� �x2 � 9

x2 � 2x � 3�

�x � 3��x � 3��x � 3��x � 1�

�x � 3

x � 1, x � 3.

f �x� � �x2 � 9��x2 � 2x � 3�.

D�x�N�x� <y � 0,

x � 2,x � �1,

�0, 0�f �0� � 0�0, 0�,

f �x� �x

�x � 1��x � 2�.

f �x� � x��x2 � x � 2�.


3

−3

−2

−1

−

1

2

3

x

Verticalasymptote:

x = 2

Verticalasymptote:

x = −1


y = 0

f(x) =x2 − x − 2

x

y

1 2

FIGURE 2.42

y

x

x2 − 9x2 − 2x − 3

f(x) =


y = 1

−3−4 1 2 3 4 5 6

2

3

−2

−3

−4

−5

Verticalasymptote:

x = −1

1

−1

FIGURE 2.43 HOLE AT x � 3


Negative

Positive

1 Negative

3 Positive �3, 0.75�f �3� � 0.75�2, ��

�1, �0.5�f �1� � �0.5�0, 2�

��0.5, 0.4�f ��0.5� � 0.4�0.5��1, 0�

��3, �0.3�f ��3� � �0.3�3��, �1�


Positive

Negative

2 Positive �2, 1.67�f �2� � 1.67��1, ��

��2, �1�f ��2� � �1�2��3, �1�

��4, �0.33�f ��4� � 0.33�4��, �3�

Example 5

If you are unsure of the shapeof a portion of the graph of arational function, plot someadditional points. Also note that when the numerator and the denominator of a rationalfunction have a common factor,the graph of the function has ahole at the zero of the commonfactor (see Example 6).

Example 6

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Slant AsymptotesConsider a rational function whose denominator is of degree 1 or greater. If thedegree of the numerator is exactly one more than the degree of the denominator,the graph of the function has a slant (or oblique) asymptote. For example, thegraph of

has a slant asymptote, as shown in Figure 2.44. To find the equation of a slantasymptote, use long division. For instance, by dividing into youobtain

Slant asymptote

As increases or decreases without bound, the remainder term approaches 0, so the graph of approaches the line as shown inFigure 2.44.

A Rational Function with a Slant Asymptote

Sketch the graph of

SolutionFactoring the numerator as allows you to recognize the -intercepts. Using long division

allows you to recognize that the line is a slant asymptote of the graph.


x-intercepts: and


Slant asymptote:

Additional points:

The graph is shown in Figure 2.45.


y � x

x � 1,

�2, 0��1, 0�

f �0� � 2�0, 2�,

y � x

f �x� �x2 � x � 2

x � 1� x �

2

x � 1

x�x � 2��x � 1�

f �x� � �x2 � x � 2��x � 1�.

y � x � 2,f2��x � 1�x

�y � x � 2�

� x � 2 �2

x � 1.f �x� �

x2 � x

x � 1

x2 � x,x � 1

f �x� �x2 � x

x � 1

Activities

1. Sketch a graph of

Answer:

2. Find the slant asymptote.

Answer: y � 4x � 2

g�x� �4x 2 � 2x � 1

x � 1

x4−8 −2

−2

2

4

6

−4

−6

y

2

f �x� �3x � 4

x2 � x � 6.

x

y = x − 2

Slantasymptote:

Verticalasymptote:

= 1x −

yf x( ) =

x + 1x x2 −

−2−4−6−8 2 4 6 8−2

−4

FIGURE 2.44

4

4

5

5

3

3

2

−2

−2

−3

−3x

Verticalasymptote:

x = 1

Slantasymptote:

y = x

f(x) =x − 1

x2 − x − 2

y

11

FIGURE 2.45

Example 7


Negative

0.5 Positive

1.5 Negative

3 Positive �3, 2�f �3� � 2�2, ��

�1.5, �2.5�f �1.5� � �2.5�1, 2�

�0.5, 4.5�f �0.5� � 4.5��1, 1�

��2, �1.33�f ��2� � �1.33�2��, �1�

333202_0206.qxd 12/7/05 9:56 AM Page 190

ApplicationsThere are many examples of asymptotic behavior in real life. For instance,Example 8 shows how a vertical asymptote can be used to analyze the cost ofremoving pollutants from smokestack emissions.

Cost-Benefit Model

A utility company burns coal to generate electricity. The cost (in dollars) ofremoving % of the smokestack pollutants is given by

for Sketch the graph of this function. You are a member of a statelegislature considering a law that would require utility companies to remove 90%of the pollutants from their smokestack emissions. The current law requires 85%removal. How much additional cost would the utility company incur as a result ofthe new law?

SolutionThe graph of this function is shown in Figure 2.46. Note that the graph has avertical asymptote at Because the current law requires 85% removal,the current cost to the utility company is

Evaluate when

If the new law increases the percent removal to 90%, the cost will be

Evaluate when

So, the new law would require the utility company to spend an additional

FIGURE 2.46


Cos

t (in

thou

sand

s of

dol

lars

)

20 40 60 80 100

200

400

600

800

1000

Percent of pollutants removed

p

C

C =80,000 p100 − p

90%

85%

Smokestack Emissions

Subtract 85% removal costfrom 90% removal cost.720,000 � 453,333 � $266,667.

p � 90.C� $720,000. C �80,000(90)

100 � 90

p � 85.C� $453,333. C �80,000(85)

100 � 85

p � 100.

0 ≤ p < 100.

C �80,000p100 � p

pC


Example 8

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If you go on to take a course in calculus, you will learn an analytic techniquefor finding the exact value of that produces a minimum area. In this case, thatvalue is x � 6�2 � 8.485.

x

Finding a Minimum Area

A rectangular page is designed to contain 48 square inches of print. The marginsat the top and bottom of the page are each 1 inch deep. The margins on each sideare inches wide. What should the dimensions of the page be so that the leastamount of paper is used?

112

Example 9

Graphical SolutionLet be the area to be minimized. From Figure 2.47,you can write

The printed area inside the margins is modeled byor To find the minimum area,

rewrite the equation for in terms of just one variableby substituting for

The graph of this rational function is shown in Figure2.48. Because represents the width of the printedarea, you need consider only the portion of the graphfor which is positive. Using a graphing utility, youcan approximate the minimum value of to occurwhen inches. The corresponding value of is

inches. So, the dimensions should be

by

FIGURE 2.48


00

24

A =(x + 3)(48 + 2x)

x , x > 0

200

y � 2 � 7.6 inches.x � 3 � 11.5 inches

48�8.5 � 5.6yx � 8.5

Ax

x

x > 0 ��x � 3��48 � 2x�

x,

A � �x � 3��48

x� 2�

y.48�xA

y � 48�x.48 � xy

A � �x � 3��y � 2�.

A

Numerical SolutionLet be the area to be minimized. From Figure 2.47, you canwrite

The printed area inside the margins is modeled by orTo find the minimum area, rewrite the equation for

in terms of just one variable by substituting for

Use the table feature of a graphing utility to create a table ofvalues for the function

beginning at From the table, you can see that the mini-mum value of occurs when is somewhere between 8 and9, as shown in Figure 2.49. To approximate the minimum valueof to one decimal place, change the table so that it starts at

and increases by 0.1. The minimum value of occurswhen as shown in Figure 2.50. The correspondingvalue of is inches. So, the dimensions shouldbe inches by inches.

FIGURE 2.49 FIGURE 2.50

y � 2 � 7.6x � 3 � 11.548�8.5 � 5.6y

x � 8.5,y1x � 8

y1

xy1

x � 1.

y1 ��x � 3��48 � 2x�

x

��x � 3��48 � 2x�

x, x > 0

A � �x � 3��48x

� 2�y.48�xA

y � 48�x.48 � xy

A � �x � 3�� y � 2�.

A

1 in.

1 in.

11 in.211 in.2y

x

FIGURE 2.47

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Exercises 2.6

In Exercises 1– 4, (a) complete each table for the function,(b) determine the vertical and horizontal asymptotes of the graph of the function, and (c) find the domain of thefunction.

1. 2.

3. 4.

In Exercises 5 –12, find the domain of the function andidentify any horizontal and vertical asymptotes.

5. 6.

7. 8.

9. 10.

11. 12.

In Exercises 13 –16, match the rational function with itsgraph. [The graphs are labeled (a), (b), (c), and (d).]

(a) (b)

(c) (d)

13. 14.

15. 16.

In Exercises 17–20, find the zeros (if any) of the rationalfunction.

17. 18.

19. 20. g�x� �x3 � 8

x 2 � 1f �x� � 1 �

3

x � 3

h�x� � 2 �5

x 2 � 2g�x� �

x 2 � 1

x � 1

f �x� � �x � 2

x � 4f �x� �

x � 1

x � 4

f �x� �1

x � 5f �x� �

2

x � 3

x

4

−4

−2−2

−4

2

y

x

4

2

−264

y

x

4

2

−2

−4

−2−4−6−8

y

x

4

2

−2

−4

6

y

f �x� �3x 2 � x � 5

x 2 � 1f �x� �

3x 2 � 1

x 2 � x � 9

f �x� �2x 2

x � 1f �x� �

x 3

x 2 � 1

f �x� �1 � 5x

1 � 2xf �x� �

2 � x

2 � x

f �x� �4

�x � 2�3f �x� �

1

x 2

x4

4

−8 8

8

y

x4−4

−4−8

−8

8

y

f �x� �4x

x2 � 1f �x� �

3x 2

x 2 � 1

x−8

−4−4 4 8

8

12

y

x−4

−4

−2 2 4−2

2

4

y

f �x� �5x

x � 1f �x� �

1

x � 1

VOCABULARY CHECK: Fill in the blanks.

1. Functions of the form where and are polynomials and is not the zero polynomial, are called ________ ________.

2. If as from the left or the right, then is a ________ ________ of the graph of

3. If as then is a ________ ________ of the graph of

4. For the rational function given by if the degree of is exactly one more than the degree of then the graph of has a ________ (or oblique) ________.

PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.

fD�x�,N�x�f �x� � N�x��D�x�,

f.y � bx → ±�,f �x� → b

f.x � ax → af �x� → ±�

D�x�D�x�N�x�f �x� � N�x��D�x�,

x

0.5

0.9

0.99

0.999

f �x� x

1.5

1.1

1.01

1.001

f �x� x

5

10

100

1000

f �x�

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In Exercises 21– 26, find the domain of the function andidentify any horizontal and vertical asymptotes.

21. 22.

23. 24.

25. 26.

In Exercises 27–46, (a) state the domain of the function, (b)identify all intercepts, (c) find any vertical and horizontalasymptotes, and (d) plot additional solution points asneeded to sketch the graph of the rational function.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39.

40.

41. 42.

43. 44.

45. 46.

Analytical, Numerical, and Graphical Analysis In Exercises47– 50, do the following.

(a) Determine the domains of and

(b) Simplify and find any vertical asymptotes of thegraph of

(c) Compare the functions by completing the table.

(d) Use a graphing utility to graph and in the sameviewing window.

(e) Explain why the graphing utility may not show thedifference in the domains of and

47.

48.

49.

50.

In Exercises 51–64, (a) state the domain of the function, (b)identify all intercepts, (c) identify any vertical and slantasymptotes, and (d) plot additional solution points asneeded to sketch the graph of the rational function.

51. 52.

53. 54.

55. 56.

57. 58.

59. 60.

61. 62. f �x� �2x 2 � 5x � 5

x � 2f �x� �

x 2 � x � 1

x � 1

g�x� �x 3

2x 2 � 8f �x� �

x 3

x 2 � 1

f �x� �x2

3x � 1f �t� � �

t 2 � 1

t � 5

h�x� �x2

x � 1g�x� �

x2 � 1x

f �x� �1 � x2

xf �x� �

2x2 � 1x

g�x� �x2 � 5

xh�x� �

x2 � 4

x

g�x� �2

x � 4f �x� �

2x � 6

x 2 � 7x � 12,

g�x� �1

xf �x� �

x � 2

x 2 � 2x,

g�x� � xf �x� �x 2�x � 2�x 2 � 2x

,

g�x� � x � 1f �x� �x 2 � 1

x � 1,

g.f

gf

f.f

g.f

f �x� �x2 � 16x � 4

f �t� �t2 � 1t � 1

f �x� �3x2 � 8x � 42x2 � 3x � 2

f �x� �2x2 � 5x � 22x2 � x � 6

f �x� �5�x � 4�

x2 � x � 12f �x� �

x2 � 3xx2 � x � 6

f �x� �x2 � x � 2

x 3 � 2x2 � 5x � 6

f �x� �2x2 � 5x � 3

x 3 � 2x2 � x � 2

g�x� �x2 � 2x � 8

x2 � 9h�x� �

x2 � 5x � 4x2 � 4

f �x� � �1

�x � 2�2g�s� �

s

s 2 � 1

f �t� �1 � 2t

tf �x� �

x 2

x 2 � 9

P�x� �1 � 3x

1 � xC�x� �

5 � 2x

1 � x

g�x� �1

3 � xh�x� �

�1

x � 2

f �x� �1

x � 3f �x� �

1

x � 2

f �x� �6x2 � 11x � 36x2 � 7x � 3

f �x� �x2 � 3x � 42x2 � x � 1

f �x� �x2 � 4

x2 � 3x � 2f �x� �

x2 � 1x2 � 2x � 3

f �x� �x � 3x2 � 9

f �x� �x � 4

x2 � 16


x 0 1 1.5 2 2.5 3

g�x�

f �x�

�1

x 0 1 2 3 4 5 6

g�x�

f �x�

x 0 0.5 1 1.5 2 3

g�x�

f �x�

�0.5

x 0 1

g�x�

f �x�

�0.5�1�1.5�2�3

333202_0206.qxd 12/7/05 9:56 AM Page 194


63.

64.

In Exercises 65– 68, use a graphing utility to graph therational function. Give the domain of the function andidentify any asymptotes. Then zoom out sufficiently far sothat the graph appears as a line. Identify the line.

65.

66.

67.

68.

Graphical Reasoning In Exercises 69–72, (a) use the graphto determine any -intercepts of the graph of the rationalfunction and (b) set and solve the resulting equationto confirm your result in part (a).

69. 70.

71. 72.

73. Pollution The cost (in millions of dollars) of removingof the industrial and municipal pollutants discharged

into a river is given by

(a) Use a graphing utility to graph the cost function.

(b) Find the costs of removing 10%, 40%, and 75% of thepollutants.

(c) According to this model, would it be possible toremove 100% of the pollutants? Explain.

74. Recycling In a pilot project, a rural township is givenrecycling bins for separating and storing recyclableproducts. The cost (in dollars) for supplying bins to of the population is given by

(a) Use a graphing utility to graph the cost function.

(b) Find the costs of supplying bins to 15%, 50%, and 90%of the population.

(c) According to this model, would it be possible to supplybins to 100% of the residents? Explain.

75. Population Growth The game commission introduces100 deer into newly acquired state game lands. The popu-lation of the herd is modeled by

where is the time in years (see figure).

(a) Find the populations when and

(b) What is the limiting size of the herd as time increases?

76. Concentration of a Mixture A 1000-liter tank contains50 liters of a 25% brine solution. You add liters of a 75%brine solution to the tank.

(a) Show that the concentration , the proportion of brineto total solution, in the final mixture is

(b) Determine the domain of the function based on thephysical constraints of the problem.

(c) Sketch a graph of the concentration function.

(d) As the tank is filled, what happens to the rate at whichthe concentration of brine is increasing? What percentdoes the concentration of brine appear to approach?

C �3x � 50

4�x � 50�.

C

x

t � 25.t � 10,t � 5,

Time (in years)

Dee

r po

pula

tion

t50 100 150 200

200

400

600

800

1000

1200

1400

N

t

t ≥ 0N �20�5 � 3t�1 � 0.04t

,

N

0 ≤ p < 100.C �25,000p

100 � p ,

p%C

0 ≤ p < 100.C �255p

100 � p ,

p%C

x−8 −4

−44 8

4

8

y

x−4

−4

−2 4

2

4

y

y � x � 3 �2

xy �

1

x� x

x−2 8642

−4

2

4

6

y

x−2 864

−4

2

4

6

y

y �2x

x � 3y �

x � 1

x � 3

y � 0x

h�x� �12 � 2x � x 2

2�4 � x�

g�x� �1 � 3x 2 � x 3

x 2

f �x� �2x 2 � x

x � 1

f �x� �x 2 � 5x � 8

x � 3

f �x� �2x3 � x2 � 8x � 4

x2 � 3x � 2

f �x� �2x3 � x2 � 2x � 1

x2 � 3x � 2

333202_0206.qxd 12/7/05 9:56 AM Page 195

77. Page Design A page that is inches wide and incheshigh contains 30 square inches of print. The top and bottommargins are 1 inch deep and the margins on each side are 2 inches wide (see figure).

(a) Show that the total area on the page is

(b) Determine the domain of the function based on thephysical constraints of the problem.

(c) Use a graphing utility to graph the area function andapproximate the page size for which the least amountof paper will be used. Verify your answer numericallyusing the table feature of the graphing utility.

78. Page Design A rectangular page is designed to contain64 square inches of print. The margins at the top andbottom of the page are each 1 inch deep. The margins oneach side are inches wide. What should the dimensionsof the page be so that the least amount of paper is used?

80. Sales The sales (in millions of dollars) for the YankeeCandle Company in the years 1998 through 2003 are shownin the table. (Source: The Yankee Candle Company)

1998 184.5 1999 256.6 2000 338.8

2001 379.8 2002 444.8 2003 508.6

A model for these data is given by

where represents the year, with corresponding to1998.

(a) Use a graphing utility to plot the data and graph themodel in the same viewing window. How well does themodel fit the data?

(b) Use the model to estimate the sales for the YankeeCandle Company in 2008.

(c) Would this model be useful for estimating sales after2008? Explain.

Synthesis

True or False? In Exercises 81 and 82, determine whetherthe statement is true or false. Justify your answer.

81. A polynomial can have infinitely many vertical asymptotes.

82. The graph of a rational function can never cross one of itsasymptotes.

Think About It In Exercises 83 and 84, write a rationalfunction that has the specified characteristics. (There aremany correct answers.)

83. Vertical asymptote: None

Horizontal asymptote:

84. Vertical asymptote:

Horizontal asymptote: None

Skills Review

In Exercises 85– 88, completely factor the expression.

85. 86.

87. 88.

In Exercises 93–96, solve the inequality and graph thesolution on the real number line.

89. 90.

91. 92.

93. Make a Decision To work an extended application analyzing the total manpower of the Department of Defense,visit this text’s website at college.hmco.com. (DataSource: U.S. Census Bureau)

12�2x � 3� ≥ 5�4�x � 2�� < 20

5 � 2x > 5�x � 1�10 � 3x ≤ 0

x 3 � 6x2 � 2x � 12x 3 � 5x2 � 4x � 20

3x2 � 23x � 36x2 � 15x � 56

x � 1x � �2,

y � 2

f

t � 8t

S �5.816t2 � 130.680.004t2 � 1.00

, 8 ≤ t ≤ 13

S

112

A �2x�x � 11�

x � 4.

A

x

y

1 in.

1 in.

2 in. 2 in.

yx

196 Chapter 2 Polynomial and Rational Functions196 Chapter 2 Polynomial and Rational Functions

79. Average Speed A driver averaged 50 miles per houron the round trip between Akron, Ohio, and Columbus,Ohio, 100 miles away. The average speeds for goingand returning were and miles per hour, respectively.

(a) Show that

(b) Determine the vertical and horizontal asymptotesof the graph of the function.

(c) Use a graphing utility to graph the function.

(d) Complete the table.

(e) Are the results in the table what you expected?Explain.

(f) Is it possible to average 20 miles per hour in onedirection and still average 50 miles per hour on theround trip? Explain.

y �25x

x � 25.

yx

Model It

x 30 35 40 45 50 55 60

y

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2.6 rational functions - ths advanced...

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