intermediate algebra chapter 12 review - miracosta...
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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.
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f x( ) = 2x
2.
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g x( ) =13
x
3.
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f(x) = 1+ 2x
4.
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f x( ) = log2 x
5.
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g x( ) = log1
2
x
6.
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g x( ) = 3x+2
7. Given
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f(x) = x and g(x) = x2 −9, find each of the following:a. f(9) + g(−1)b. f(x) + g(x)c.
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f(25)•g(x)d.
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(f o g)(5)e.
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(g o f)(16)
8. Find the equation of the inverse of the given one-to-one functions:
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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.
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log32 2 =15
b.
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log3 81= 4
c.
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log51
25= −2
10. Write each of the following as a logarithmic equation.
a.
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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.
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log5 1
15.
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6log6 5
16.
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log25 5
17.
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log4
116
18.
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log7 719.
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log1000
20.
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log5 56
21.
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lne22.
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log3 81
23.
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log 1024.
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eln300
25.
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log2 32
26.
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10log 8x( )
Solve each equation algebraically without using a calculator. Giveexact answers.27.
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log4 x = 328.
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logx 16 = 429.
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log9 3 = x
30.
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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.
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log2
32x
33.
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ln5
e2
34.
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log525
y3
35.
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logbx2
z + 136.
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log3 x2 2x −1( )
Write as a single logarithm. Don’t use a calculator; but, wherepossible, evaluate logarithm expressions without a calculator.37.
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3logb x + 2logb y
38.
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15
lnx + 2lny
39.
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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.
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4x = 1642.
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4x = 32
43.
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4x =14
44.
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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.
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e−1.25
47.
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log2 1748.
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log0.5 2.149.
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log5.150.
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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.
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10x = 8.753.
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e2x = 6.154.
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20−2.1x = 055.
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103x−1 = 3.756.
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4x+1 = 5x
57.
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log2 3x + 1( ) = 7
58.
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log(2x −3) = 3.259.
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lnx = −360.
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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
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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.
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x + x2 −9
17. −2
7c.
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5x2 −457d. 47e. 7
18.
€
12
8a.
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f−1(x) = 5x −2 19. 3
8b.
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f−1(x) = x3 + 4
8c.
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f−1(x) =x3 −1
2
20. 6
9a.
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321/ 5 = 2 21. 1
9b.
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34 = 81
9c.
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5−2 =1
25
22. 4
10a.
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log3181
= −4
10b.
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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 =
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log3log5
≈ 0.6826
32.
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5− log2 x52.x =
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log8.7log10
=log8.7
1≈ 0.9395
33.
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ln5−253. x =
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log6.12loge
≈ 0.9041
34.
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2−3log5 y35.
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2logb x − logb(z + 1)54. x =
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log20log2.1
≈ 4.0377
36.
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2log3 x + log3 2x −1( )55. x =
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1+log3.7log103
≈ 0.5227
37.
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logb x3y2( )38.
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ln y2 x5( )56. x =
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−log4log4 − log5
≈ 6.2126
39.
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logb
x + 1( )6
y3
40.
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log2x3 y + 1
y
57. x =
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27 −13
≈ 42.3333
58.
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x =103.2 + 3
2=793.9466
59. x =
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e−3≈ 0.049841. x = 2
60. x =
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519
≈ 0.2632
42.
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x =52
61. x = 2
43. x = −1 62. x =
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e2 −4≈ 3.3891
44.
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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.
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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