Intermediate AlgebraChapter 12 Review

Set up a Table of Coordinates and graph the given functions. Find they-intercept. Label at least three points on the graph. Your graph musthave the correct shape. For each exponential function, give theequation of the horizontal asymptote.1.

€

f x( ) = 2x

2.

€

g x( ) =13

x

3.

€

f(x) = 1+ 2x

4.

€

f x( ) = log2 x

5.

€

g x( ) = log1

2

x

6.

€

g x( ) = 3x+2

7. Given

€

f(x) = x and g(x) = x2 −9, find each of the following:a. f(9) + g(−1)b. f(x) + g(x)c.

€

f(25)•g(x)d.

€

(f o g)(5)e.

€

(g o f)(16)

8. Find the equation of the inverse of the given one-to-one functions:

€

a. f(x) =x + 2

5

b. f(x) = (x −4)3

c. f(x) = 2x + 13

9. Write each of the following as an exponential equation.

a.

€

log32 2 =15

b.

€

log3 81= 4

c.

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log51

25= −2

10. Write each of the following as a logarithmic equation.

a.

€

3−4 =181

b.

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251/ 2 = 5

11. Graph the inverse of the given one-to-one function on the sameset of axes. Label at least three points on the graph.

12. What test do you use to determine if a given graph represents afunction that has an inverse function? State the test.

13. Determine whether each graph represents a one-to one function(If so, it the function has an inverse function). State your conclusion.a.

b.

c.

Find the value of each of the following without using a calculator.Leave your answers in exact form (no logarithms in any answer).14.

€

log5 1

15.

€

6log6 5

16.

€

log25 5

17.

€

log4

116

18.

€

log7 719.

€

log1000

20.

€

log5 56

21.

€

lne22.

€

log3 81

23.

€

log 1024.

€

eln300

25.

€

log2 32

26.

€

10log 8x( )

Solve each equation algebraically without using a calculator. Giveexact answers.27.

€

log4 x = 328.

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logx 16 = 429.

€

log9 3 = x

30.

€

log3181

= x

31. State the product rule, quotient rule, and power rule forlogarithms. (See section 5, chapter 12.)

Write each expression as a sum or difference of logarithms. Don’tuse a calculator; but, where possible, evaluate logarithm expressions.(Example: If the answer contains

€

log5 25, simplify to obtain 2).

32.

€

log2

32x

33.

€

ln5

e2

34.

€

log525

y3

35.

€

logbx2

z + 136.

€

log3 x2 2x −1( )

Write as a single logarithm. Don’t use a calculator; but, wherepossible, evaluate logarithm expressions without a calculator.37.

€

3logb x + 2logb y

38.

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15

lnx + 2lny

39.

€

6logb x + 1( ) −3logb y

40.

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3log2 x +12

log2 y + 1( ) − log2 y

Solve each equation algebraically. Do not use a calculator, and giveall answers in exact form (no logarithms allowed).41.

€

4x = 1642.

€

4x = 32

43.

€

4x =14

44.

€

25x = 545.

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62x+1 = 36

Use a calculator to approximate the following to four decimal places.(Hint: The change-of-base formula may be needed on a few ofthese.)46.

€

e−1.25

47.

€

log2 1748.

€

log0.5 2.149.

€

log5.150.

€

ln4.8

Solve each equation by using an appropriate method. After you havegotten an exact answer, use your calculator to approximate theanswer to the nearest thousandth (three decimal places).51.

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5x = 352.

€

10x = 8.753.

€

e2x = 6.154.

€

20−2.1x = 055.

€

103x−1 = 3.756.

€

4x+1 = 5x

57.

€

log2 3x + 1( ) = 7

58.

€

log(2x −3) = 3.259.

€

lnx = −360.

€

log5 x − log5 4x −1( ) = 1

61.

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log4 x + 2( ) − log4 x −1( ) = 1

62.

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ln x + 4 = 1

Solve each of the following application problems algebraically. Foreach problem, you may use your calculator but must show enoughwork to outline your solution strategy.63. How much will an investment of $15,000 be worth in 30 years ifthe annual interest rate is 5.5% and compounding is

a. quarterly?b. monthly?c. continuously?

64. The equation

€

y = 158.97(1.012)x models the population of theUnited States from 1950 through 2003. In this equation, y is thepopulation in millions and x represents the number of years after1950. Use the equation to estimate the population of the UnitedStates in 1975. (Use a calculator and round your answer to twodecimal places).

Answers:1-6. See graphs on following pages.

16.

€

12

7a. –5

7b.

€

x + x2 −9

17. −2

7c.

€

5x2 −457d. 47e. 7

18.

€

12

8a.

€

f−1(x) = 5x −2 19. 3

8b.

€

f−1(x) = x3 + 4

8c.

€

f−1(x) =x3 −1

2

20. 6

9a.

€

321/ 5 = 2 21. 1

9b.

€

34 = 81

9c.

€

5−2 =1

25

22. 4

10a.

€

log3181

= −4

10b.

€

log25 5 =12

11. See graph on following pages.

23.

€

12

12. Use the horizontal line test.HLT: A function f is one-to-one andthus has an inverse function if thereis no horizontal line that intersectsthe graph of the function f at morethan one point.

24. 300

13a. Yes, the function has aninverse.

25. 5

13b. No, the function does nothave an inverse because it doesnot pass the horizontal line test.

26. 8x

13c. Yes, the function has aninverse—it passes the horizontalline test.

27. x = 64

14. 0 28. x = 215. 5 29. x = 1/2

30. −4 50. 1.568631. See chapter 12, section5.

51. x =

€

log3log5

≈ 0.6826

32.

€

5− log2 x52.x =

€

log8.7log10

=log8.7

1≈ 0.9395

33.

€

ln5−253. x =

€

log6.12loge

≈ 0.9041

34.

€

2−3log5 y35.

€

2logb x − logb(z + 1)54. x =

€

log20log2.1

≈ 4.0377

36.

€

2log3 x + log3 2x −1( )55. x =

€

1+log3.7log103

≈ 0.5227

37.

€

logb x3y2( )38.

€

ln y2 x5( )56. x =

€

−log4log4 − log5

≈ 6.2126

39.

€

logb

x + 1( )6

y3

40.

€

log2x3 y + 1

y

57. x =

€

27 −13

≈ 42.3333

58.

€

x =103.2 + 3

2=793.9466

59. x =

€

e−3≈ 0.049841. x = 2

60. x =

€

519

≈ 0.2632

42.

€

x =52

61. x = 2

43. x = −1 62. x =

€

e2 −4≈ 3.3891

44.

€

x =12

63a. $77,231.65

45.

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x =12

63b. $77,810.8263c. $78, 104.70

46. 0.2865 64. 214.20 million people47. 4.087548. −1.070449. 0.7076

1.

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f x( ) = 2x, Horizontal Asymptote is the x-axis or y = 0 (y = 0 is the

equation of the x-axis) Y-intercept is (1, 0) x y

-1 1/201

12

2.

€

g x( ) =13

x

, H.A. y = 0 (y = 0 is the equation of the x-axis)

Y-intercept is (1, 0) x y

-1 301

11/3

3.

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f(x) = 1+ 2x. H.A. y = 1. Y-intercept is (0. 2) x y

-1 3/201

23

4.

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f x( ) = log2 x, There is no y-intercept. The graph does not cross the

y-axis. x y

1/2 -112

01

5.

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g x( ) = log1

2

x . Note that the graph will not cross the y-axis.

X y2 -111/2

01

6. g(x) =

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3x+2. Shift left two units. Horizontal asymptote does notmove—it remains y = 0. Y-intercept is (0, 9).

x g (x)-3-2-1

1/313

11.