Name: Class: Date: ID: A PreCalculus: Semester 1 Final Exam Review Short Answer 1. Determine whether the relation represents a function. If it is a function, state the domain and range. 9. Find the domain of the function. f(x) = x 2 + 4 10. Find the domain of the function. h(x) = 2. Determine whether the relation represents a function. If it is a function, state the domain and range. {(19, -4), (3, -3), (3, 0), (12, 3), (28, 5)} 3. Determine whether the equation defines y as a function of x. y = x 4. Determine whether the equation defines y as a function of x. y 2 = 6 - x 2 5. Find the value for the function. Find f(-9) when f(x) = x - 6. 6. Find the value for the function. 11. For the given functions f and g, find the requested function and state its domain. f(x) = 4x - 3; g(x) = 8x - 9 Find f - g. 12. For the given functions f and g, find the requested function and state its domain. f(x) = 3x + 4; g(x) = 4x - 6 Find f g. 13. Solve the problem. Find (f + g)(-2) when f(x) = x - 3 and g(x) = x + 1. 14. Solve the problem. Find (-3) when f(x) = 3x - 4 and g(x) = 3x 2 + 14x + 3. Find f(4) when f(x) =. 7. Find the value for the function. Find -f(x) when f(x) = 2x 2-3x + 4. 8. Find the value for the function. Find f(x - 1) when f(x) = 3x 2-5x - 5. 1
Name: ID: A 15. Determine whether the graph is that of a function. If it is, use the graph to find its domain and range, the intercepts, if any, and any symmetry with respect to the x-axis, the y-axis, or the origin. 17. The graph of a function f is given. Use the graph to answer the question. Use the graph of f given below to find f(-6). 16. Determine whether the graph is that of a function. If it is, use the graph to find its domain and range, the intercepts, if any, and any symmetry with respect to the x-axis, the y-axis, or the origin. 18. The graph of a function f is given. Use the graph to answer the question. For what numbers x is f(x) = 0? 19. Answer the question about the given function. Given the function f(x) = -2x 2-4x - 8, is the point (-1, -6) on the graph of f? 2
Name: ID: A 20. Answer the question about the given function. Given the function f(x) =, is the point (-2, 8) 23. The graph of a function is given. Decide whether it is even, odd, or neither. on the graph of f? 21. The graph of a function is given. Decide whether it is even, odd, or neither. 24. Determine algebraically whether the function is even, odd, or neither. f(x) = -2x 4 - x 2 22. The graph of a function is given. Decide whether it is even, odd, or neither. 25. Determine algebraically whether the function is even, odd, or neither. f(x) = -5x 2-4 26. Graph the function. f(x) = 3
Name: ID: A 27. Graph the function. f(x) = 31. Graph the function by starting with the graph of the basic function and then using the techniques of shifting, compressing, stretching, and/or reflecting. f(x) = (x - 7) 2 + 4 28. Match the correct function to the graph. 32. Use U = universal set = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 5, 8}, B = {2, 3, 5, 7}, and C = {1, 4, 9} to find the set. A C 33. Use U = universal set = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 5, 8}, B = {2, 3, 5, 7}, and C = {1, 4, 9} to find the set. A B 29. Write an equation that results in the indicated translation. The absolute value function, shifted 5 units to the left 30. Write an equation that results in the indicated translation. The square root function, shifted 5 units upward 34. Use U = universal set = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 5, 8}, B = {2, 3, 5, 7}, and C = {1, 4, 9} to find the set. (A B) C 35. Use U = universal set = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 5, 8}, B = {2, 3, 5, 7}, and C = {1, 4, 9} to find the set. 36. Evaluate the expression using the given values. -3xy + 8y - 5 x = 4, y = 3 4
Name: ID: A 37. Evaluate the expression using the given values. x = 7, y = 8 38. Simplify the expression. Express the answer so that all exponents are positive. Whenever an exponent is 0 or negative, we assume that the base is not 0. (-4x 2 ) -1 39. Simplify the expression. Express the answer so that all exponents are positive. Whenever an exponent is 0 or negative, we assume that the base is not 0. (x 9 y -1 ) 3 40. Simplify the expression. Express the answer so that all exponents are positive. Whenever an exponent is 0 or negative, we assume that the base is not 0. (x -6 y 6 ) -7 z 9 41. Simplify the expression. Express the answer so that all exponents are positive. Whenever an exponent is 0 or negative, we assume that the base is not 0. -1 42. Tell whether the expression is a polynomial. If it is, give its degree. 7x 2-43. Tell whether the expression is a polynomial. If it is, give its degree. 7z 6 + z 44. Add, subtract, or multiply, as indicated. Express your answer as a single polynomial in standard form. 8(1 - y 3 ) + 5(1 + y + y 2 + y 3 ) 45. Multiply the polynomials using the special product formulas. Express the answer as a single polynomial in standard form. (2x - 10)(2x + 10) 46. Multiply the polynomials using the special product formulas. Express the answer as a single polynomial in standard form. (x - 10) 2 47. Find the quotient and the remainder. 9x 8-15x 4 divided by 3x 48. Find the quotient and the remainder. 6x 2 + 17x - 28 divided by x + 4 49. Find the quotient and the remainder. x 4 + 6x 2 + 7 divided by x 2 + 1 50. Factor completely. If the polynomial cannot be factored, say it is prime. 9x 2-1 51. Factor completely. If the polynomial cannot be factored, say it is prime. 27y 3-1 52. Factor completely. If the polynomial cannot be factored, say it is prime. x 2 + 2x + 1 5
Name: ID: A 53. Factor completely. If the polynomial cannot be factored, say it is prime. 81x 2-126x + 49 62. Evaluate the expression using the values given in the table. (g f)(1) 54. Factor completely. If the polynomial cannot be factored, say it is prime. 2x 2-2x - 12 55. Factor completely. If the polynomial cannot be factored, say it is prime. 10x 2 + 21x + 9 63. Evaluate the expression using the values given in the table. 56. Use synthetic division to find the quotient and the remainder. x 5 + x 2-4 is divided by x + 3 57. Use synthetic division to find the quotient and the remainder. -3x 3-9x 2 + 10x - 8 is divided by x + 4 58. Use synthetic division to determine whether x - c is a factor of the given polynomial. x 3-4x 2-39x + 126; x + 6 59. Use synthetic division to determine whether x - c is a factor of the given polynomial. x 3-9x 2 + 8x + 64; x + 6 60. Reduce the rational expression to lowest terms. f(g(-5)) 64. For the given functions f and g, find the requested composite function value. f(x) =, g(x) = 5x; Find (f g)(3). 65. 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). 61. Reduce the rational expression to lowest terms. 66. For the given functions f and g, find the requested composite function value. f(x) = 3x + 8, g(x) = ; Find (g f)(3). 6
Name: ID: A 67. For the given functions f and g, find the requested composite function. 74. Determine whether the function is one-to-one. f(x) =, g(x) = ; Find (f g)(x). 68. Decide whether the composite functions, f g and g f, are equal to x. f(x) = x 2 + 1, g(x) = - 1 69. Decide whether the composite functions, f g and g f, are equal to x. f(x) =, g(x) = x 2 75. Indicate whether the function is one-to-one. {(-20, -18), (10, -18), (-8, 18)} 76. Use the horizontal line test to determine whether the function is one-to-one. 70. Find functions f and g so that f g = H. H(x) = 71. Find functions f and g so that f g = H. H(x) = 72. Find the domain of the composite function f g. f(x) = x + 9; g(x) = 77. Use the horizontal line test to determine whether the function is one-to-one. 73. Find the domain of the composite function f g. f(x) = ; g(x) = 7
Name: ID: A 78. Find the inverse of the function and state its domain and range. {(-4, -8), (8, 4), (-3, -2), (3, 2)} 79. Graph the function as a solid line or curve and its inverse as a dashed line or curve on the same axes. 2y - 10 = 4x 82. Decide whether or not the functions are inverses of each other. f(x) = (x - 4) 2, x 4; g(x) = + 4 83. The function f is one-to-one. Find its inverse. f(x) = 6x 2-3, x 0 84. The function f is one-to-one. Find its inverse. f(x) = 85. The function f is one-to-one. State the domain and the range of f and f -1. f(x) = 80. Graph the function as a solid line or curve and its inverse as a dashed line or curve on the same axes. f(x) = 86. State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. f(x) = 87. State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. f(x) = 11 88. Form a polynomial whose zeros and degree are given. Zeros: -3, -2, 2; degree 3 81. Decide whether or not the functions are inverses of each other. f(x) = 3x + 9, g(x) = x - 3 89. Form a polynomial whose zeros and degree are given. Zeros: 0, - 3, 2; degree 3 8
Name: ID: A 90. For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. 97. Use the graph to determine the domain and range of the function. f(x) = 4(x - 7)(x - 5) 4 91. 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) = 3(x + 2)(x - 4) 3 92. Find the x- and y-intercepts of f. f(x) = 2x 3 (x - 2) 5 93. Find the x- and y-intercepts of f. f(x) = (x + 2)(x - 5)(x + 5) 94. Find the domain of the rational function. 98. Graph the function using transformations. f(x) = + 1 G(x) = 95. Find the domain of the rational function. R(x) = 96. Use the graph to determine the domain and range of the function. 9
Name: ID: A 99. Graph the function using transformations. f(x) = + 1 104. Graph the function. f(x) = 100. Find the vertical asymptotes of the rational function. f(x) = 105. Graph the function. f(x) = 101. Find the vertical asymptotes of the rational function. f(x) = 102. Give the equation of the horizontal asymptote, if any, of the function. h(x) = 103. Give the equation of the oblique asymptote, if any, of the function. h(x) = 106. Solve the inequality algebraically. Express the solution in interval notation. (x - 5) 2 (x + 7) > 0 107. Solve the inequality algebraically. Express the solution in interval notation. (x + 2)(x - 2)(x - 7) < 0 10
Name: ID: A Essay 108. Analyze the graph of the given function f as follows: (a) Determine the end behavior. (b) Find the x and y intercepts of the graph. (c) Determine whether the graph crosses or touches the x-axis at each x-intercept. (d) Find the domain of f. (f) Use the information obtained in (a) (d) to draw a complete graph of f by hand. 109. Analyze the graph of the given function f as follows: (a) Determine the end behavior. (b) Find the x and y intercepts of the graph. (c) Determine whether the graph crosses or touches the x-axis at each x-intercept. (d) Find the domain of f. (f) Use the information obtained in (a) (d) to draw a complete graph of f by hand. 11
PreCalculus: Semester 1 Final Exam Review Answer Section SHORT ANSWER 1. ANS: function domain: {Alice, Brad, Carl} range: {cat, dog} 2. ANS: not a function 3. ANS: function 4. ANS: not a function 5. ANS: 3 6. ANS: 4 7. ANS: -2x 2 + 3x - 4 8. ANS: 3x 2-11x + 3 9. ANS: all real numbers 10. ANS: {x x -4, 0, 4} 1
11. ANS: (f - g)(x) = -4x + 6; all real numbers 12. ANS: (f g)(x) = 12x 2-2x - 24; all real numbers 13. ANS: -6 14. ANS: 15. ANS: not a function 16. ANS: function domain: {x x > 0} range: all real numbers intercept: (1, 0) symmetry: none 17. ANS: 0 18. ANS: -3, 3.5, 5 19. ANS: Yes 20. ANS: No 2
21. ANS: even 22. ANS: neither 23. ANS: odd 24. ANS: even 25. ANS: even 26. ANS: 3
27. ANS: 28. ANS: y = 29. ANS: y = 30. ANS: y = + 5 31. ANS: 32. ANS: {1, 2, 3, 4, 5, 8, 9} 4
33. ANS: {2, 3, 5} 34. ANS: {1, 2, 3, 4, 5, 9} 35. ANS: {0, 2, 3, 4, 5, 6, 7, 8, 9} 36. ANS: -17 37. ANS: 38. ANS: - 39. ANS: 40. ANS: 41. ANS: 42. ANS: Not a polynomial 5
43. ANS: Polynomial; degree 6 44. ANS: -3y 3 + 5y 2 + 5y + 13 45. ANS: 4x 2-100 46. ANS: x 2-20x + 100 47. ANS: 3x 7-5x 3 ; remainder 0 48. ANS: 6x - 7; remainder 0 49. ANS: x 2 + 5; remainder 2 50. ANS: (3x - 1)(3x + 1) 51. ANS: (3y - 1)(9y 2 + 3y + 1) 52. ANS: (x + 1) 2 53. ANS: (9x - 7) 2 54. ANS: 2(x + 2)(x - 3) 6
55. ANS: (2x + 3)(5x + 3) 56. ANS: x 4-3x 3 + 9x 2-26x + 78; remainder -238 57. ANS: -3x 2 + 3x - 2; remainder 0 58. ANS: Yes 59. ANS: No 60. ANS: 61. ANS: 4x - 9 62. ANS: -8 63. ANS: 1 64. ANS: 3 65. ANS: 1937 7
66. ANS: - 67. ANS: 68. ANS: No, no 69. ANS: No, no 70. ANS: f(x) = ; g(x) = x 2-4 71. ANS: f(x) = ; g(x) = 5x + 4 72. ANS: {x 73. ANS: {x 74. ANS: One-to-one 75. ANS: No 8
76. ANS: No 77. ANS: Yes 78. ANS: {(-8, -4), (4, 8), (-2, -3), (2, 3)} D = {-8, 4, -2, 2}; R = {-4, 8, -3, 3} 79. ANS: 80. ANS: 81. ANS: Yes 9
82. ANS: Yes 83. ANS: f -1 (x) = 84. ANS: f -1 (x) = 85. ANS: f(x): D = {x x 2}, R = {y 0}; f -1 (x): D = {x x 0}, R = {y y 2} 86. ANS: Yes; degree 3 87. ANS: Yes; degree 0 88. ANS: f(x) = x 3 + 3x 2-4x - 12 for a = 1 89. ANS: f(x) = x 3 + x 2-6x for a = 1 90. ANS: 7, multiplicity 1, crosses x-axis; 5, multiplicity 4, touches x-axis 91. ANS: -2, multiplicity 1, crosses x-axis; 4, multiplicity 3, crosses x-axis 92. ANS: x-intercepts: 0, 2; y-intercept: 0 10
93. ANS: x-intercepts: -2, -5, 5; y-intercept: -50 94. ANS: all real numbers 95. ANS: {x x -9, 4} 96. ANS: domain: {x x 2} range: {y y 3} 97. ANS: domain: {x x 0} range: all real numbers 98. ANS: 11
99. ANS: 100. ANS: x = -3, x = 3 101. ANS: x =, x = -1 102. ANS: none 103. ANS: none 12
104. ANS: 105. ANS: 106. ANS: (-, -7) 107. ANS: (-, -2) (2, 7) 13
ESSAY 108. ANS: 14
109. ANS: 15