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Pre-Calculus Final Exam Review Units 1 - 3
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the value for the function.
Find f(x - 1) when f(x) = 3x2 - 5x - 5.
A. 3x2 - 11x - 7 C. -11x2 + 3x + 3
B. 3x2 - 20x - 7 D. 3x2 - 11x + 3
____ 2. Find the domain of the function.
f(x) =
A. {x|x > -7} C. {x|x 0}
B. all real numbers D. {x|x -7}
____ 3. Find the domain of the function.
g(x) =
A. {x|x 0} C. {x|x -3, 3}
B. {x|x > 9} D. all real numbers
____ 4. Find the domain of the function.
A. {x|x 5} C. all real numbers
B. {x|x 5} D. {x|x > 5}
____ 5. For the given functions f and g, find the requested function and state its domain.
f(x) = ; g(x) = 4x - 7
Find .
A. ( )(x) = ; {x|x 0}
C. ( )(x) = ; {x|x 0}
B. ( )(x) = ; {x|x }
D. ( )(x) = ; {x|x 0, x }
____ 6. Determine algebraically whether the function is even, odd, or neither.
f(x) =
A. neither C. even
B. odd
____ 7. Determine algebraically whether the function is even, odd, or neither.
f(x) =
A. even C. odd
B. neither
____ 8. Determine algebraically whether the function is even, odd, or neither.
f(x) =
A. even C. odd
B. neither
____ 9. Use a graphing utility to graph the function over the indicated interval and approximate any local
maxima and local minima. Determine where the function is increasing and where it is decreasing. If
necessary, round answers to two decimal places.
f(x) = x3 - 3x2 + 3, (-2, 3)
A. local maximum at (1, 1)
local minimum at (-1, 5)
increasing on (-2, -1)
decreasing on (-1, 1)
B. local maximum at (0, 3)
local minimum at (2, -1)
increasing on (-2, 0) and (2, 3)
decreasing on (0, 2)
C. local maximum at (-1, 5)
local minimum at (1, 1)
increasing on (-1, 1)
decreasing on (-2, -1) and (1, 2)
D. local maximum at (0, 3)
local minimum at (2, -1)
increasing on (-1, 1)
decreasing on (-2, -1) and (1, 3)
____ 10. Graph the function.
f(x) =
A.
C.
B.
D.
____ 11. Graph the function.
f(x) =
A.
C.
B.
D.
____ 12. Suppose the point (2, 4) is on the graph of y = f(x). Find a point on the graph of the given function.
y = f(x + 3)
A. (2, 7) C. (2, 1)
B. (5, 4) D. (-1, 4)
____ 13. Suppose the point (2, 4) is on the graph of y = f(x). Find a point on the graph of the given function.
f(x) + 6
A. (-4, 4) C. (8, 4)
B. (2, 10) D. (2, -6)
____ 14. For the given functions f and g, find the requested composite function value.
f(x) = 4x + 6, g(x) = 4x2 + 1; Find (g f)(4).
A. 16,901 C. 266
B. 1937 D. 94
____ 15. For the given functions f and g, find the requested composite function.
f(x) = -2x + 7, g(x) = 3x + 5; Find (g f)(x).
A. -6x + 17 C. -6x + 26
B. -6x - 16 D. 6x + 26
____ 16. For the given functions f and g, find the requested composite function.
f(x) = 4x2 + 2x + 5, g(x) = 2x - 6; Find (g f)(x).
A. 4x2 + 2x - 1 C. 4x2 + 4x + 4
B. 8x2+ 4x + 16 D. 8x2 + 4x + 4
____ 17. Find functions f and g so that f g = H.
H(x) =
A. f(x) = ; g(x) = x + 1
C. f(x) = ; g(x) = x + 1
B. f(x) = x + 1 ; g(x) =
D. f(x) = ; g(x) = 1
____ 18. Find functions f and g so that f g = H.
H(x) =
A. f(x) = ; g(x) = - 9
C. f(x) = x2 - 5; g(x) =
B. f(x) = , g(x) = x2 - 5 D. f(x) = - 9; g(x) =
____ 19. Find the domain of the composite function f g.
f(x) = ; g(x) = x + 3
A. {x C. {x
B. {x D. {x
____ 20. Find the domain of the composite function f g.
f(x) = 2x + 8; g(x) =
A. {x C. {x
B. {x D. {x
____ 21. The function f is one-to-one. Find its inverse.
f(x) = 5x + 7
A. f-1(x) = C. f-1(x) =
B. f(x) = D. f-1(x) = -
____ 22. The function f is one-to-one. Find its inverse.
f(x) = (x + 2)3 - 8.
A. f-1(x) =
C. f-1(x) = - 2
B. f-1(x) =
D. f-1(x) = + 8
____ 23. Form a polynomial whose zeros and degree are given.
Zeros: 1, multiplicity 2; 5, multiplicity 1; degree 3
A. x3 + 7x2 + 11x + 5 C. x3 + 7x2 + 10x + 5
B. x3 - 7x2 + 11x - 5 D. x3 - 2x2 + 11x - 5
____ 24. For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or
touches the x-axis at each x -intercept.
f(x) = 4(x - 7)(x - 5)4
A. -7, multiplicity 1, crosses x-axis; -5, multiplicity 4, touches x-axis
B. -7, multiplicity 1, touches x-axis; -5, multiplicity 4, crosses x-axis
C. 7, multiplicity 1, touches x-axis; 5, multiplicity 4, crosses x-axis
D. 7, multiplicity 1, crosses x-axis; 5, multiplicity 4, touches x-axis
____ 25. For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or
touches the x-axis at each x -intercept.
f(x) = x4(x2 - 5)
A. 0, multiplicity 4, touches x-axis;
, multiplicity 1, crosses x-axis;
- , multiplicity 1, crosses x-axis
B. 0, multiplicity 4, crosses x-axis;
, multiplicity 1, touches x-axis;
- , multiplicity 1, touches x-axis
C. 0, multiplicity 4, touches x-axis
D. 0, multiplicity 4, crosses x-axis
____ 26. Solve the problem.
Which of the following polynomial functions might have the graph shown in the illustration below?
A. f(x) = x(x - 2)2(x - 1) C. f(x) = x(x - 2)(x - 1)2
B. f(x) = x2(x - 2)2(x - 1)2 D. f(x) = x2(x - 2)(x - 1)
____ 27. Find the x- and y-intercepts of f.
f(x) = (x - 2)(x - 5)
A. x-intercepts: -2, -5; y-intercept: 10 C. x-intercepts: 2, 5; y-intercept: -7
B. x-intercepts: -2, -5; y-intercept: -7 D. x-intercepts: 2, 5; y-intercept: 10
____ 28. Solve the problem.
The amount of water (in gallons) in a leaky bathtub is given in the table below. Using a graphing utility, fit
the data to a third degree polynomial (or a cubic). Then approximate the time at which there is maximum
amount of water in the tub, and estimate the time when the water runs out of the tub. Express all your
answers rounded to two decimal places.
A. maximum amount of water after 5.30 minutes; water never runs out
B. maximum amount of water after 5.37 minutes; water runs out after 11.06 minutes
C. maximum amount of water after 5.30 minutes; water runs out after 8.23 minutes
D. maximum amount of water after 8.23 minutes; water runs out after 19.73 minutes
____ 29. Find the domain of the rational function.
F(x) = .
A. {x|x 3, x -3, x -5} C. {x|x 3, x -5}
B. {x|x -3, x 5} D. all real numbers
____ 30. Use the graph to determine the domain and range of the function.
A. domain: {x|x 0}
range: all real numbers
B. domain: all real numbers
range: {y|y -2 or y 2}
C. domain: {x|x 0}
range: {y|y -2 or y 2}
D. domain: {x|x -2 or x 2}
range: {y|y 0}
____ 31. Find the vertical asymptotes of the rational function.
F(x) =
A. x = 1, x = -4 C. x = -1, x = 4
B. x = -1 D. x = -1, x = -4
____ 32. Give the equation of the horizontal asymptote, if any, of the function.
H(x) =
A. y = 7 C. y = 0
B. none D. y = 8
____ 33. Give the equation of the horizontal asymptote, if any, of the function.
F(x) =
A. y = 4 C. y = 3
B. y = 1 D. none
____ 34. Give the equation of the oblique asymptote, if any, of the function.
F(x) =
A. x = y + 10 C. y = x - 6
B. y = x + 10 D. none
____ 35. Give the equation of the oblique asymptote, if any, of the function.
R(x) =
A. none C. y = x + 9
B. y = 9x D. y = 0
____ 36. Solve the problem.
Decide which of the rational functions might have the given graph.
A. y =
C. y =
B. y =
D. y =
____ 37. Solve the problem.
Determine which rational function R(x) has a graph that crosses the x-axis at -1, touches the x-axis at -4, has
vertical asymptotes at x = -2 and x = 3, and has one horizontal asymptote at y = -2.
A. R(x) = , x -2, 3 C. R(x) = , x -4, -1
B. R(x) = , x 2, -3
D. R(x) = , x -2, 3
____ 38. Use the graph of the function f to solve the inequality.
f(x) 0
A. (- , -5) (2, 7) C. (- , -5] [2, 7]
B. (- , -5] [2, ) D. (- , -5) (2, )
____ 39. Use the graph of the function f to solve the inequality.
f(x) < 0
A. (-5, 1) (6, ) C. [-5, 1] [6, )
B. (- , -5) (1, 6) D. (6, )
____ 40. Solve the inequality algebraically. Express the solution in interval notation.
(x - 5)2(x + 7) > 0
A. (- , -7) C. (- , -7]
B. (-7, ) D. (- , -7) (7, )
____ 41. Solve the inequality algebraically. Express the solution in interval notation.
(x + 7)(x + 6)(x - 6) > 0
A. (- , -7) (-6, 6) C. (6, )
B. (-7, -6) (6, ) D. (- , -6)
____ 42. Solve the equation.
4(3x - 5 ) = 256
A. {-3} C. {128}
B. {3} D.
____ 43. Change the exponential expression to an equivalent expression involving a logarithm.
72 = 49
A. log497 = 2 C. log249 = 7
B. log72 = 49 D. log749 = 2
____ 44. Change the exponential expression to an equivalent expression involving a logarithm.
ex = 6
A. log6 x = e C. logx e = 6
B. ln 6 = x D. ln x = 6
____ 45. Express as a single logarithm.
9ln (x - 3) - 11 ln x
A. ln 99x(x - 3) C. ln x11(x - 3)9
B. ln D. ln
____ 46. Solve the equation.
log2(3x - 2) - log2(x - 5) = 4
A. {6} C.
B. {18} D.
____ 47. Solve the problem.
What principal invested at 8% compounded continuously for 4 years will yield $1190? Round the answer to
two decimal places.
A. $1188.62 C. $864.12
B. $627.48 D. $1638.78
____ 48. Solve the problem.
How long does it take $1125 to triple if it is invested at 7% interest, compounded quarterly? Round your
answer to the nearest tenth.
A. 18.1 mo C. 15.8 mo
B. 18.1 yr D. 15.8 yr
____ 49. Solve the problem.
The size P of a small herbivore population at time t (in years) obeys the function if they have
enough food and the predator population stays constant. After how many years will the population reach
3000?
A. 18.64 yr C. 22.7 yr
B. 11.5 yr D. 55.59 yr
____ 50. Solve the problem.
A certain radioactive isotope has a half-life of 555 years. Determine the annual decay rate, k.
A. 0.135% C. 0.125%
B. 0.195% D. 0.265%
____ 51. Solve the problem.
The logistic growth model represents the population of a bacterium in a culture tube
after t hours. What was the initial amount of bacteria in the population?
A. 50 C. 45
B. 46 D. 44
____ 52. Solve the problem.
The logistic growth model represents the population of a bacterium in a culture tube
after t hours. When will the amount of bacteria be 630?
A. 11.26 hr C. 2.16 hr
B. 5.22 hr D. 8.2 hr
____ 53. Solve the problem.
A biologist has a bacteria sample. She records the amount of bacteria every week for 8 weeks and finds that
the exponential function of best fit to the data is A = 150 • 1.79t. Express the function of best fit in the form
A. 87.33e1.79t C. A = 150e0.58t
B. A = 0.58e150t D. A = 87.33e0.58t
____ 54. Solve the problem.
A life insurance company uses the following rate table for annual premiums for women for term life
insurance. Use a graphing utility to fit an exponential function to the data. Predict the annual premium for a
woman aged 70 years.
A. y = 0.0000398x4.06, $1233 C. y = 8.94e0.068x, $1044
B. y = 6.367e0.068x, $743 D. y = -9306.4 + 2516.3 ln (x), $1723
____ 55. Solve the problem.
Use the data in the table to build a logistic model for the population of the city t years after 1900.
A. y =
C. y =
B. y =
D. y =
____ 56. Use the TABLE feature of a graphing utility to find the limit.
(x2 + 8x - 2)
A. 0 C. 18
B. -18 D. does not exist
____ 57. Use the graph shown to determine if the limit exists. If it does, find its value.
f(x)
A. does not exist C. 1
B. 4 D. -1
____ 58. Use the graph shown to determine if the limit exists. If it does, find its value.
f(x)
A. 2 C. 1
B. 0 D. does not exist
____ 59. Use the graph shown to determine if the limit exists. If it does, find its value.
f(x)
A. does not exist C. 5
B. 3 D. 4
____ 60. Use a graphing utility to find the indicated limit rounded to two decimal places.
A. 3.00 C. 2.96
B. 2.04 D. 2.00
____ 61. Find the limit algebraically.
(x3 + 5x2 - 7x + 1)
A. 0 C. 29
B. does not exist D. 15
____ 62. Find the limit algebraically.
A. does not exist C. 0
B. 4 D. -4
____ 63. Find the numbers at which f is continuous. At which numbers is f discontinuous?
f(x) = 4x - 5
A. continuous for all real numbers except x =
B. continuous for all real numbers except x = -
C. continuous for all real numbers
D. continuous for all real numbers except x = 5
____ 64. Find the numbers at which f is continuous. At which numbers is f discontinuous?
f(x) =
A. continuous for all real numbers except x = 2
B. continuous for all real numbers except x = -2 and x = 2
C. continuous for all real numbers
D. continuous for all real numbers except x = -2, x = 2 and x = -
____ 65. Find the numbers at which f is continuous. At which numbers is f discontinuous?
f(x) =
A. continuous for all real numbers except x = -5, x = 8, and x = 3
B. continuous for all real numbers except x = 8, x = 3
C. continuous for all real numbers except x = -8, x = -3
D. continuous for all real numbers except x = 5, x = -8, and x = -3
Pre-Calculus Final Exam Review Units 1 - 3
Answer Section
MULTIPLE CHOICE
1. ANS: D PTS: 1
2. ANS: B PTS: 1
3. ANS: C PTS: 1
4. ANS: D PTS: 1
5. ANS: D PTS: 1
6. ANS: A PTS: 1
7. ANS: A PTS: 1
8. ANS: C PTS: 1
9. ANS: B PTS: 1
10. ANS: D PTS: 1
11. ANS: A PTS: 1
12. ANS: D PTS: 1
13. ANS: B PTS: 1
14. ANS: B PTS: 1
15. ANS: C PTS: 1
16. ANS: D PTS: 1
17. ANS: A PTS: 1
18. ANS: B PTS: 1
19. ANS: C PTS: 1
20. ANS: C PTS: 1
21. ANS: A PTS: 1
22. ANS: C PTS: 1
23. ANS: B PTS: 1
24. ANS: D PTS: 1
25. ANS: A PTS: 1
26. ANS: C PTS: 1
27. ANS: D PTS: 1
28. ANS: C PTS: 1
29. ANS: C PTS: 1
30. ANS: C PTS: 1
31. ANS: B PTS: 1
32. ANS: A PTS: 1
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38. ANS: C PTS: 1
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47. ANS: C PTS: 1
48. ANS: D PTS: 1
49. ANS: B PTS: 1
50. ANS: C PTS: 1
51. ANS: C PTS: 1
52. ANS: A PTS: 1
53. ANS: C PTS: 1
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55. ANS: A PTS: 1
56. ANS: C PTS: 1
57. ANS: D PTS: 1
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63. ANS: C PTS: 1
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65. ANS: B PTS: 1
