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E. Applications Involving Absolute Value

E. Applications Involving Absolute Value

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College Algebra G&M—



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Section 2.3 Absolute Value Functions, Equations, and Inequalities



EXAMPLE 8







225



Solving Applications Involving Absolute Value Inequalities

For new cars, the number of miles per gallon (mpg) a car will get is heavily

dependent on whether it is used mainly for short trips and city driving, or primarily

on the highway for longer trips. For a certain car, the number of miles per gallon

that a driver can expect varies by no more than 6.5 mpg above or below its field

tested average of 28.4 mpg. What range of mileage values can a driver expect

for this car?



Solution







Field tested average: 28.4 mpg

mileage varies by no more than 6.5 mpg

Ϫ6.5



gather information

highlight key phrases



ϩ6.5



28.4



make the problem visual



Let m represent the miles per gallon a driver can expect.

Then the difference between m and 28.4 can be no more

than 6.5, or Ϳm Ϫ 28.4Ϳ Յ 6.5.

Ϳm Ϫ 28.4Ϳ Յ 6.5

Ϫ6.5 Յ m Ϫ 28.4 Յ 6.5

21.9 Յ m Յ 34.9



assign a variable

write an equation model

equation model

apply Property I

add 28.4 to all three parts



The mileage that a driver can expect ranges from a low of 21.9 mpg

to a high of 34.9 mpg.

E. You’ve just seen how

we can solve applications

involving absolute value



Now try Exercises 61 through 70







2.3 EXERCISES





CONCEPTS AND VOCABULARY



Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.



1. When multiplying or dividing by a negative

quantity, we

the inequality symbol to

maintain a true statement.



2. To write an absolute value equation or inequality in

simplified form, we

the absolute value

expression on one side.



3. The absolute value equation Ϳ2x ϩ 3Ϳ ϭ 7 is true

when 2x ϩ 3 ϭ

or when 2x ϩ 3 ϭ

.



4. The absolute value inequality Ϳ3x Ϫ 6Ϳ 6 12 is

true when 3x Ϫ 6 7

and 3x Ϫ 6 6



Describe the solution set for each inequality (assume k > 0). Justify your answer.



5. Ϳax ϩ bͿ 6 Ϫk



6. Ϳax ϩ bͿ 7 Ϫk



.



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CHAPTER 2 More on Functions



DEVELOPING YOUR SKILLS



Solve each absolute value equation. Write the solution in

set notation. For Exercises 7 to 18, verify solutions by

substituting into the original equation. For Exercises

19–26 verify solutions using a calculator.



7. 2Ϳm Ϫ 1Ϳ Ϫ 7 ϭ 3

8. 3Ϳn Ϫ 5Ϳ Ϫ 14 ϭ Ϫ2

9. Ϫ3Ϳx ϩ 5Ϳ ϩ 6 ϭ Ϫ15

10. Ϫ2Ϳy ϩ 3Ϳ Ϫ 4 ϭ Ϫ14

11. 2Ϳ4v ϩ 5Ϳ Ϫ 6.5 ϭ 10.3

12. 7Ϳ2w ϩ 5Ϳ ϩ 6.3 ϭ 11.2

13. ϪͿ7p Ϫ 3Ϳ ϩ 6 ϭ Ϫ5

14. ϪͿ3q ϩ 4Ϳ ϩ 3 ϭ Ϫ5

15. Ϫ2ͿbͿ Ϫ 3 ϭ Ϫ4

16. Ϫ3ͿcͿ Ϫ 5 ϭ Ϫ6

17. Ϫ2Ϳ3xͿ Ϫ 17 ϭ Ϫ5

18. Ϫ5Ϳ2yͿ Ϫ 14 ϭ 6

19. Ϫ3 `



w

ϩ 4 ` Ϫ 1 ϭ Ϫ4

2



20. Ϫ2 ` 3 Ϫ



v

` ϩ 1 ϭ Ϫ5

3



21. 8.7Ϳp Ϫ 7.5Ϳ Ϫ 26.6 ϭ 8.2



35.



Ϳ5v ϩ 1Ϳ

ϩ8 6 9

4



36.



Ϳ3w Ϫ 2Ϳ

ϩ6 6 8

2



37. `



1

7

4x ϩ 5

Ϫ ` Յ

3

2

6



38. `



2y Ϫ 3

3

15

Ϫ ` Յ

4

8

16



39. Ϳn ϩ 3Ϳ 7 7

40. Ϳm Ϫ 1Ϳ 7 5

41. Ϫ2ͿwͿ Ϫ 5 Յ Ϫ11

42. Ϫ5ͿvͿ Ϫ 3 Յ Ϫ23

43.

44.



ͿqͿ

2

ͿpͿ

5



Ϫ



5

1

Ն

6

3



ϩ



3

9

Ն

2

4



45. 3Ϳ5 Ϫ 7dͿ ϩ 9 Ն 15

46. 5Ϳ2c ϩ 7Ϳ ϩ 1 Ն 11

47. 2 6 ` Ϫ3m ϩ



1

4

` Ϫ

5

5



3

5

Ϫ 2n ` Ϫ

4

4



22. 5.3Ϳq ϩ 9.2Ϳ ϩ 6.7 ϭ 43.8



48. 4 Յ `



23. 8.7ͿϪ2.5xͿ Ϫ 26.6 ϭ 8.2



49. 4Ϳ5 Ϫ 2hͿ Ϫ 9 7 11



24. 5.3Ϳ1.25nͿ ϩ 6.7 ϭ 43.8



50. 3Ϳ7 ϩ 2kͿ Ϫ 11 7 10



25. Ϳx Ϫ 2Ϳ ϭ Ϳ3x ϩ 4Ϳ



51. 3.9Ϳ4q Ϫ 5Ϳ ϩ 8.7 Յ Ϫ22.5



26. Ϳ2x Ϫ 1Ϳ ϭ Ϳx ϩ 3Ϳ



52. 0.9Ϳ2p ϩ 7Ϳ ϩ 16.11 Յ 10.89



Solve each absolute value inequality. Write solutions in

interval notation. Check solutions by back substitution,

or using a calculator.



27. 3Ϳp ϩ 4Ϳ ϩ 5 6 8

28. 5Ϳq Ϫ 2Ϳ Ϫ 7 Յ 8

29. Ϫ3 ͿmͿ Ϫ 2 7 4

30. Ϫ2ͿnͿ ϩ 3 7 7

31. Ϳ3b Ϫ 11Ϳ ϩ 6 Յ 9

32. Ϳ2c ϩ 3Ϳ Ϫ 5 6 1

33. Ϳ4 Ϫ 3zͿ ϩ 12 6 7

34. Ϳ2 Ϫ 3uͿ ϩ 5 Յ 4



53. Ϳ4z Ϫ 9Ϳ ϩ 6 Ն 4

54. Ϳ5u Ϫ 3Ϳ ϩ 8 7 6

Use the intersect command on a graphing calculator

and the given functions to solve (a) f 1x2 ϭ g1x2 ,

(b) f 1x2 Ն g1x2 , and (c) f 1x2 6 g1x2 .



55. f 1x2 ϭ Ϳx Ϫ 3Ϳ ϩ 2, g1x2 ϭ 12x ϩ 2



56. f 1x2 ϭ ϪͿx ϩ 2Ϳ Ϫ 1, g1x2 ϭ Ϫ32x Ϫ 9



57. f 1x2 ϭ 0.5Ϳx ϩ 3Ϳ ϩ 1, g1x2 ϭ Ϫ2Ϳx ϩ 1Ϳ ϩ 5

58. f 1x2 ϭ 2Ϳx Ϫ 3Ϳ ϩ 2, g1x2 ϭ Ϳx Ϫ 4Ϳ ϩ 6



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Section 2.3 Absolute Value Functions, Equations, and Inequalities



WORKING WITH FORMULAS



59. Spring Oscillation: Ϳd Ϫ xͿ Յ L

A weight attached to a spring hangs at rest a distance

of x in. off the ground. If the weight is pulled down

(stretched) a distance of L inches and released, the

weight begins to bounce and its distance d off the

ground must satisfy the indicated formula. (a) If x

equals 4 ft and the spring is stretched 3 in. and

released, solve the inequality to find what distances

from the ground the weight will oscillate between.

(b) Solve for x in terms of L and d.





227



60. A “Fair” Coin: `



h Ϫ 50

` 6 1.645

5



If we flipped a coin 100 times, we expect “heads” to

come up about 50 times if the coin is “fair.” In a study

of probability, it can be shown that the number of

heads h that appears in such an experiment should

satisfy the given inequality to be considered “fair.”

(a) Solve this inequality for h. (b) If you flipped a

coin 100 times and obtained 40 heads, is the coin

“fair”?



APPLICATIONS



Solve each application of absolute value.



61. Altitude of jet stream: To take advantage of the jet

stream, an airplane must fly at a height h (in feet)

that satisfies the inequality Ϳh Ϫ 35,050Ϳ Յ 2550.

Solve the inequality and determine if an altitude of

34,000 ft will place the plane in the jet stream.

62. Quality control tests: In order to satisfy quality

control, the marble columns a company produces

must earn a stress test score S that satisfies the

inequality ͿS Ϫ 17,750Ϳ Յ 275. Solve the inequality

and determine if a score of 17,500 is in the passing

range.

63. Submarine depth: The sonar operator on a

submarine detects an old World War II submarine

net and must decide to detour over or under the

net. The computer gives him a depth model

Ϳd Ϫ 394Ϳ Ϫ 20 7 164, where d is the depth in feet

that represents safe passage. At what depth should the

submarine travel to go under or over the net? Answer

using simple inequalities.

64. Optimal fishing depth: When deep-sea fishing,

the optimal depths d (in feet) for catching a

certain type of fish satisfy the inequality

28Ϳd Ϫ 350Ϳ Ϫ 1400 6 0. Find the range of depths

that offer the best fishing. Answer using simple

inequalities.

For Exercises 65 through 68, (a) develop a model that

uses an absolute value inequality, and (b) solve.



65. Stock value: My stock in MMM Corporation

fluctuated a great deal in 2009, but never by more

than $3.35 from its current value. If the stock is

worth $37.58 today, what was its range in 2009?



66. Traffic studies: On a

given day, the volume

of traffic at a busy

intersection averages

726 cars per hour

(cph). During rush hour

the volume is much

higher, during “off

hours” much lower.

Find the range of this

volume if it never

varies by more than

235 cph from the

average.

67. Physical training for recruits: For all recruits in the

3rd Armored Battalion, the average number of sit-ups

is 125. For an individual recruit, the amount varies

by no more than 23 sit-ups from the battalion

average. Find the range of sit-ups for this battalion.

68. Computer consultant salaries: The national

average salary for a computer consultant is

$53,336. For a large computer firm, the salaries

offered to their employees vary by no more than

$11,994 from this national average. Find the range

of salaries offered by this company.

69. Tolerances for sport balls: According to the official

rules for golf, baseball, pool, and bowling, (a) golf

balls must be within 0.03 mm of d ϭ 42.7 mm,

(b) baseballs must be within 1.01 mm of

d ϭ 73.78 mm, (c) billiard balls must be within

0.127 mm of d ϭ 57.150 mm, and (d) bowling balls

must be within 12.05 mm of d ϭ 2171.05 mm. Write

each statement using an absolute value inequality,

then (e) determine which sport gives the least

width of interval

b for the diameter

tolerance t at ϭ

average value

of the ball.



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70. Automated packaging: The machines that fill

boxes of breakfast cereal are programmed to fill

each box within a certain tolerance. If the box is

overfilled, the company loses money. If it is

underfilled, it is considered unsuitable for sale.





Suppose that boxes marked “14 ounces” of cereal

must be filled to within 0.1 oz. Find the acceptable

range of weights for this cereal.



EXTENDING THE CONCEPT



71. Determine the value or values (if any) that will

make the equation or inequality true.

x

a. ͿxͿ ϩ x ϭ 8

b. Ϳx Ϫ 2Ϳ Յ

2

c. x Ϫ ͿxͿ ϭ x ϩ ͿxͿ

d. Ϳx ϩ 3Ϳ Ն 6x

e. Ϳ2x ϩ 1Ϳ ϭ x Ϫ 3

72. The equation Ϳ5 Ϫ 2xͿ ϭ Ϳ3 ϩ 2xͿ has only one

solution. Find it and explain why there is only one.

73. In many cases, it can be helpful to view the solutions

to absolute value equations and inequalities as

follows. For any algebraic expression X and positive







2–42



CHAPTER 2 More on Functions



constant k, the equation ͿXͿ ϭ k has solutions X ϭ k

and ϪX ϭ k, since the absolute value of either

quantity on the left will indeed yield the positive

constant k. Likewise, ͿXͿ 6 k has solutions X 6 k

and ϪX 6 k. Note the inequality symbol has not

been reversed as yet, but will naturally be reversed

as part of the solution process. Solve the following

equations or inequalities using this idea.

a. Ϳx Ϫ 3Ϳ ϭ 5

b. Ϳx Ϫ 7Ϳ 7 4

c. 3Ϳx ϩ 2Ϳ Յ 12

d. Ϫ3Ϳx Ϫ 4Ϳ ϩ 7 ϭ Ϫ11



MAINTAINING YOUR SKILLS



74. (R.4) Factor the expression completely:

18x3 ϩ 21x2 Ϫ 60x.

76. (R.7) Simplify



Ϫ1



by rationalizing the

3 ϩ 23

denominator. State the result in exact form and

approximate form (to hundredths).



75. (1.5) Solve V2 ϭ



2W

for ␳ (physics).

C␳A



77. (R.3) Solve the inequality, then write the solution

set in interval notation:

Ϫ312x Ϫ 52 7 21x ϩ 12 Ϫ 7.



MID-CHAPTER CHECK

1. Determine whether the following function is even,

ͿxͿ

odd, or neither. f 1x2 ϭ x2 ϩ

4x

2. Use a graphing calculator to find the maximum and

minimum values of

f 1x2 ϭ Ϫ1.91x4 Ϫ 2.3x3 ϩ 2.2x Ϫ 5.12 . Round to

the nearest hundredth.

3. Use interval notation to identify the interval(s)

where the function from Exercise 2 is increasing,

decreasing, or constant. Round to the nearest

hundredth.



4. Write the equation of the function that has the same

graph of f 1x2 ϭ 2x, shifted left 4 units and up 2

units.

5. For the graph given, (a) identify

the function family, (b) describe

or identify the end-behavior,

inflection point, and x- and

y-intercepts, (c) determine

the domain and range, and

(d) determine the value of k if

f 1k2 ϭ 2.5. Assume required

features have integer values.



Exercise 5

y

5



f(x)



Ϫ5



5 x



Ϫ5



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Reinforcing Basic Concepts



6. Use a graphing calculator to graph the given

functions in the same window and comment on

what you observe.

p1x2 ϭ 1x Ϫ 32 2



r1x2 ϭ Ϫ12 1x Ϫ 32 2



229



9. Solve the following absolute value inequalities.

Write solutions in interval notation.

a. 3.1Ϳd Ϫ 2Ϳ ϩ 1.1 Ն 7.3

Ϳ1 Ϫ yͿ

11

ϩ2 7

b.

3

2

c. Ϫ5Ϳk Ϫ 2Ϳ ϩ 3 6 4



q1x2 ϭ Ϫ1x Ϫ 32 2



7. Solve the following absolute value equations. Write

the solution in set notation.

2

11

a. Ϳd Ϫ 5Ϳ ϩ 1 ϭ 7

b. 5 Ϫ Ϳs ϩ 3Ϳ ϭ

3

2



10. Kiteboarding: With the correct sized kite, a person

can kiteboard when the wind is blowing at a speed

w (in mph) that satisfies the inequality Ϳw Ϫ 17Ϳ Յ 9.

Solve the inequality and determine if a person can

kiteboard with a windspeed of (a) 5 mph?

(b) 12 mph?



8. Solve the following absolute value inequalities.

Write solutions in interval notation.

x

a. 3Ϳq ϩ 4Ϳ Ϫ 2 6 10 b. ` ϩ 2 ` ϩ 5 Յ 5

3



REINFORCING BASIC CONCEPTS

Using Distance to Understand Absolute Value Equations and Inequalities

For any two numbers a and b on the number line, the distance between a and b can be written Ϳa Ϫ bͿ or Ϳb Ϫ aͿ. In exactly

the same way, the equation Ϳx Ϫ 3Ϳ ϭ 4 can be read, “the distance between 3 and an unknown number is equal to 4.”

The advantage of reading it in this way (instead of “the absolute value of x minus 3 is 4”), is that a much clearer

visualization is formed, giving a constant reminder there are two solutions. In diagram form we have Figure 2.51.

Distance between

3 and x is 4.

Ϫ5 Ϫ4 Ϫ3 Ϫ2



Figure 2.51



4 units

Ϫ1



0



1



4 units

2



3



4



5



Distance between

3 and x is 4.

6



7



8



9



From this we note the solutions are x ϭ Ϫ1 and x ϭ 7.

In the case of an inequality such as Ϳx ϩ 2Ϳ Յ 3, we rewrite the inequality as Ϳx Ϫ 1Ϫ22 Ϳ Յ 3 and read it, “the distance

between Ϫ2 and an unknown number is less than or equal to 3.” With some practice, visualizing this relationship

mentally enables a quick statement of the solution: x ʦ 3Ϫ5, 14 . In diagram form we have Figure 2.52.

Distance between Ϫ2

and x is less than or equal to 3.

Ϫ8 Ϫ7 Ϫ6



Figure 2.52



3 units



Distance between Ϫ2

and x is less than or equal to 3.



3 units



Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1



0



1



2



3



4



5



6



Equations and inequalities where the coefficient of x is not 1 still lend themselves to this form of conceptual understanding. For Ϳ2x Ϫ 1Ϳ Ն 3 we read, “the distance between 1 and twice an unknown number is greater than or equal to 3.”

On the number line (Figure 2.53), the number 3 units to the right of 1 is 4, and the number 3 units to the left of 1 is Ϫ2.

Distance between 1 and

2x is greater than or equal to 3.



Figure 2.53



3 units



Ϫ6 Ϫ5 Ϫ4 Ϫ3



؊2 Ϫ1



0



3 units

1



2



3



Distance between 1 and

2x is greater than or equal to 3

4



5



6



7



8



For 2x Յ Ϫ2, x Յ Ϫ1, and for 2x Ն 4, x Ն 2, and the solution set is x ʦ 1Ϫq, Ϫ1 4 ´ 32, q 2.

Attempt to solve the following equations and inequalities by visualizing a number line. Check all results algebraically.

Exercise 1: Ϳx Ϫ 2Ϳ ϭ 5



Exercise 2: Ϳx ϩ 1Ϳ Յ 4



Exercise 3: Ϳ2x Ϫ 3Ϳ Ն 5



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2.4



Basic Rational Functions and Power Functions;

More on the Domain



LEARNING OBJECTIVES

In Section 2.4 you will see

how we can:



A. Graph basic rational

functions, identify vertical

and horizontal

asymptotes, and describe

end-behavior

B. Use transformations to

graph basic rational

functions and write the

equation for a given

graph

C. Graph basic power

functions and state their

domains

D. Solve applications

involving basic rational

and power functions



In this section, we introduce two new kinds of relations, rational functions and power

functions. While we’ve already studied a variety of functions, we still lack the ability

to model a large number of important situations. For example, functions that model the

amount of medication remaining in the bloodstream over time, the relationship

between altitude and weightlessness, and the equations modeling planetary motion

come from these two families.



A. Rational Functions and Asymptotes

Just as a rational number is the ratio of two integers, a rational function is the ratio of

two polynomials. In general,

Rational Functions

A rational function V(x) is one of the form

V1x2 ϭ



p1x2

d1x2



,



where p and d are polynomials and d1x2 0.

The domain of V(x) is all real numbers, except the zeroes of d.

The simplest rational functions are the reciprocal function y ϭ 1x and the reciprocal square function y ϭ x12, as both have a constant numerator and a single term in

the denominator. Since division by zero is undefined, the domain of both excludes

x ϭ 0. A preliminary study of these two functions will provide a strong foundation for

our study of general rational functions in Chapter 4.



The Reciprocal Function: y ‫؍‬



1

x



The reciprocal function takes any input (other than zero) and gives its reciprocal as the

output. This means large inputs produce small outputs and vice versa. A table of

values (Table 2.1) and the resulting graph (Figure 2.54) are shown.

Table 2.1



230



Figure 2.54



x



y



x



y



Ϫ1000



Ϫ1/1000



1/1000



1000



Ϫ5



Ϫ1/5



1/3



3



Ϫ4



Ϫ1/4



1/2



2



Ϫ3



Ϫ1/3



1



1



Ϫ2



Ϫ1/2



2



1/2



Ϫ1



Ϫ1



3



1/3



Ϫ1/2



Ϫ2



4



1/4



Ϫ1/3



Ϫ3



5



1/5



Ϫ1/1000



Ϫ1000



1000



1/1000



0



undefined



y

3



΂a, 3΃







1

x



2



(1, 1)



΂Ϫ3, Ϫ a΃



1



Ϫ5



΂3, a΃

5



΂Ϫ5, Ϫ Q΃

(Ϫ1, Ϫ1)



΂Ϫ a, Ϫ3΃



΂5, Q΃

x



Ϫ1

Ϫ2

Ϫ3



2–44



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Section 2.4 Basic Rational Functions and Power Functions; More on the Domain



WORTHY OF NOTE

The notation used for graphical

behavior always begins by

describing what is happening to the

x-values, and the resulting effect on

the y-values. Using Figure 2.55,

visualize that for a point (x, y) on

the graph of y ϭ 1x , as x gets larger,

y must become smaller, particularly

since their product must always be

1 1y ϭ 1x 1 xy ϭ 12 .



Table 2.1 and Figure 2.54 reveal some interesting features. First, the graph passes

the vertical line test, verifying y ϭ 1x is indeed a function. Second, since division by

zero is undefined, there can be no corresponding point on the graph, creating a break

at x ϭ 0. In line with our definition of rational functions, the domain is

x ʦ 1Ϫq, 02 ´ 10, q 2 . Third, this is an odd function, with a “branch” of the graph in

the first quadrant and one in the third quadrant, as the reciprocal of any input maintains

its sign. Finally, we note in QI that as x becomes an infinitely large positive number, y

becomes infinitely small and closer to zero. It seems convenient to symbolize this endbehavior using the following notation:



Figure 2.55



as x S q,



y



yS0



as x becomes an infinitely

large positive number



Graphically, the curve becomes very close to, or approaches the x-axis.

We also note that as x approaches zero from the right, y becomes an infinitely large

positive number: as x S 0 ϩ , y S q . Note a superscript ϩ or Ϫ sign is used to indicate the direction of the approach, meaning from the positive side (right) or from the

negative side (left).



y

x



x



EXAMPLE 1



y approaches 0







Describing the End-Behavior of Rational Functions

For y ϭ 1x in QIII (Figure 2.54),

a. Describe the end-behavior of the graph.

b. Describe what happens as x approaches zero.



Solution







Similar to the graph’s behavior in QI, we have

a. In words: As x becomes an infinitely large negative number, y approaches zero.

In notation: As x S Ϫq , y S 0.

b. In words: As x approaches zero from the left, y becomes an infinitely large

negative number. In notation: As x S 0 Ϫ , y S Ϫq .

Now try Exercises 7 and 8



The Reciprocal Square Function: y ‫؍‬







1

x2



From our previous work, we anticipate this graph will also have a break at x ϭ 0. But

since the square of any negative number is positive, the branches of the reciprocal

square function are both above the x-axis. Note the result is the graph of an even function. See Table 2.2 and Figure 2.56.

Table 2.2



Figure 2.56



x



y



x



y



Ϫ1000



1/1,000,000



1/1000



1,000,000



Ϫ5



1/25



1/3



9



Ϫ4



1/16



1/2



4



Ϫ3



1/9



1



1



Ϫ2



1/4



2



1/4



Ϫ1



1



3



1/9



Ϫ1/2



4



4



1/16



Ϫ1/3



9



5



1/25



Ϫ1/1000



1,000,000



1000



1/1,000,000



0



undefined



y ϭ x12



y

3



(Ϫ1, 1)



΂Ϫ5,



1



1

25 ΃

Ϫ5



2



΂Ϫ3, 19΃



(1, 1)



΂3, 19 ΃



΂5,

5



Ϫ1

Ϫ2

Ϫ3



1

25 ΃



x



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E. Applications Involving Absolute Value

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