Pre-Calculus Final Exam Review Units 1-3

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1 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) = 3x 2-5x - 5. A. 3x 2-11x - 7 C. -11x 2 + 3x + 3 B. 3x 2-20x - 7 D. 3x 2-11x 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

2 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) = x 3-3x 2 + 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)

3 10. Graph the function. f(x) = A. C. B. D.

4 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)

5 14. For the given functions f and g, find the requested composite function value. f(x) = 4x + 6, g(x) = 4x 2 + 1; Find (g f)(4). A. 16,901 C. 266 B D 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 For the given functions f and g, find the requested composite function. f(x) = 4x 2 + 2x + 5, g(x) = 2x - 6; Find (g f)(x). A. 4x 2 + 2x - 1 C. 4x 2 + 4x + 4 B. 8x 2 + 4x + 16 D. 8x 2 + 4x 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) = Find functions f and g so that f g = H. H(x) = A. f(x) = ; g(x) = - 9 C. f(x) = x 2-5; g(x) = B. f(x) =, g(x) = x 2-5 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

6 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) = 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) = Form a polynomial whose zeros and degree are given. Zeros: 1, multiplicity 2; 5, multiplicity 1; degree 3 A. x 3 + 7x x + 5 C. x 3 + 7x x + 5 B. x 3-7x x - 5 D. x 3-2x x 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) = x 4 (x 2-5) 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

7 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) = x 2 (x - 2) 2 (x - 1) 2 D. f(x) = x 2 (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: 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 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 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

8 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 = Give the equation of the horizontal asymptote, if any, of the function. H(x) = A. y = 7 C. y = 0 B. none D. y = 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

9 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 = 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, 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, )

10 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. 7 2 = 49 A. log 497 = 2 C. log 249 = 7 B. log 72 = 49 D. log 749 = Change the exponential expression to an equivalent expression involving a logarithm. e x = 6 A. log 6 x = e C. log x e = 6 B. ln 6 = x D. ln x = Express as a single logarithm. 9ln (x - 3) - 11 ln x A. ln 99x(x - 3) C. ln x 11 (x - 3) 9 B. ln D. ln

11 46. Solve the equation. log 2(3x - 2) - log 2(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. $ C. $ B. $ D. $ 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 mo C mo B yr D 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 yr C yr B yr D yr 50. Solve the problem. A certain radioactive isotope has a half-life of 555 years. Determine the annual decay rate, k. A % C % B % D % 51. Solve the problem. The logistic growth model after t hours. What was the initial amount of bacteria in the population? A. 50 C. 45 B. 46 D Solve the problem. represents the population of a bacterium in a culture tube 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 hr C hr B hr D. 8.2 hr

12 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 = t. Express the function of best fit in the form A e 1.79t C. A = 150e 0.58t B. A = 0.58e 150t D. A = 87.33e 0.58t 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 = x 4.06, $1233 C. y = 8.94e 0.068x, $1044 B. y = 6.367e 0.068x, $743 D. y = ln (x), $ Solve the problem. Use the data in the table to build a logistic model for the population of the city t years after A. y = C. y = B. y = D. y = 56. Use the TABLE feature of a graphing utility to find the limit. (x 2 + 8x - 2) A. 0 C. 18 B. -18 D. does not exist

13 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 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

14 60. Use a graphing utility to find the indicated limit rounded to two decimal places. A C B D Find the limit algebraically. (x 3 + 5x 2-7x + 1) A. 0 C. 29 B. does not exist D Find the limit algebraically. A. does not exist C. 0 B. 4 D 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 = 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 = 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

15 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: ANS: D PTS: ANS: A PTS: ANS: D PTS: ANS: B PTS: ANS: B PTS: ANS: C PTS: ANS: D PTS: ANS: A PTS: ANS: B PTS: ANS: C PTS: ANS: C PTS: ANS: A PTS: ANS: C PTS: ANS: B PTS: ANS: D PTS: ANS: A PTS: ANS: C PTS: ANS: D PTS: ANS: C PTS: ANS: C PTS: ANS: C PTS: ANS: B PTS: ANS: A PTS: ANS: B PTS: ANS: B PTS: ANS: A PTS: ANS: B PTS: ANS: D PTS: ANS: C PTS: ANS: A PTS: ANS: A PTS: ANS: B PTS: 1

16 42. ANS: B PTS: ANS: D PTS: ANS: B PTS: ANS: D PTS: ANS: A PTS: ANS: C PTS: ANS: D PTS: ANS: B PTS: ANS: C PTS: ANS: C PTS: ANS: A PTS: ANS: C PTS: ANS: C PTS: ANS: A PTS: ANS: C PTS: ANS: D PTS: ANS: A PTS: ANS: A PTS: ANS: D PTS: ANS: D PTS: ANS: D PTS: ANS: C PTS: ANS: B PTS: ANS: B PTS: 1

Evaluate the expression using the values given in the table. 1) (f g)(6) x f(x) x g(x)

M60 (Precalculus) ch5 practice test Evaluate the expression using the values given in the table. 1) (f g)(6) 1) x 1 4 8 1 f(x) -4 8 0 15 x -5-4 1 6 g(x) 1-5 4 8 For the given functions f and g, find the