PreCalculus: Semester 1 Final Exam Review


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1 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 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 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 Solve the problem. Find (3) when f(x) = 3x  4 and g(x) = 3x x + 3. Find f(4) when f(x) =. 7. Find the value for the function. Find f(x) when f(x) = 2x 23x Find the value for the function. Find f(x  1) when f(x) = 3x 25x
2 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 xaxis, the yaxis, 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 xaxis, the yaxis, 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 24x  8, is the point (1, 6) on the graph of f? 2
3 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 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 Graph the function. f(x) = 3
4 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) 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
5 Name: ID: A 37. Evaluate the expression using the given values. x = 7, y = 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 ) 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 ) 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 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 Tell whether the expression is a polynomial. If it is, give its degree. 7x 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) Find the quotient and the remainder. 9x 815x 4 divided by 3x 48. Find the quotient and the remainder. 6x x  28 divided by x Find the quotient and the remainder. x 4 + 6x divided by x Factor completely. If the polynomial cannot be factored, say it is prime. 9x Factor completely. If the polynomial cannot be factored, say it is prime. 27y Factor completely. If the polynomial cannot be factored, say it is prime. x 2 + 2x + 1 5
6 Name: ID: A 53. Factor completely. If the polynomial cannot be factored, say it is prime. 81x 2126x 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 22x Factor completely. If the polynomial cannot be factored, say it is prime. 10x x Evaluate the expression using the values given in the table. 56. Use synthetic division to find the quotient and the remainder. x 5 + x 24 is divided by x Use synthetic division to find the quotient and the remainder. 3x 39x x  8 is divided by x Use synthetic division to determine whether x  c is a factor of the given polynomial. x 34x 239x + 126; x Use synthetic division to determine whether x  c is a factor of the given polynomial. x 39x 2 + 8x + 64; x 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
7 Name: ID: A 67. For the given functions f and g, find the requested composite function. 74. Determine whether the function is onetoone. 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) = Decide whether the composite functions, f g and g f, are equal to x. f(x) =, g(x) = x Indicate whether the function is onetoone. {(20, 18), (10, 18), (8, 18)} 76. Use the horizontal line test to determine whether the function is onetoone. 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 onetoone. 73. Find the domain of the composite function f g. f(x) = ; g(x) = 7
8 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) = The function f is onetoone. Find its inverse. f(x) = 6x 23, x The function f is onetoone. Find its inverse. f(x) = 85. The function f is onetoone. 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) = Form a polynomial whose zeros and degree are given. Zeros: 3, 2, 2; degree Decide whether or not the functions are inverses of each other. f(x) = 3x + 9, g(x) = x Form a polynomial whose zeros and degree are given. Zeros: 0,  3, 2; degree 3 8
9 Name: ID: A 90. For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the xaxis at each x intercept. 97. Use the graph to determine the domain and range of the function. f(x) = 4(x  7)(x  5) For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the xaxis at each x intercept. f(x) = 3(x + 2)(x  4) Find the x and yintercepts of f. f(x) = 2x 3 (x  2) Find the x and yintercepts 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
10 Name: ID: A 99. Graph the function using transformations. f(x) = 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) > Solve the inequality algebraically. Express the solution in interval notation. (x + 2)(x  2)(x  7) < 0 10
11 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 xaxis at each xintercept. (d) Find the domain of f. (f) Use the information obtained in (a) (d) to draw a complete graph of f by hand 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 xaxis at each xintercept. (d) Find the domain of f. (f) Use the information obtained in (a) (d) to draw a complete graph of f by hand. 11
12 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 ANS: 3x 211x ANS: all real numbers 10. ANS: {x x 4, 0, 4} 1
13 11. ANS: (f  g)(x) = 4x + 6; all real numbers 12. ANS: (f g)(x) = 12x 22x  24; all real numbers 13. ANS: ANS: 15. ANS: not a function 16. ANS: function domain: {x x > 0} range: all real numbers intercept: (1, 0) symmetry: none 17. ANS: ANS: 3, 3.5, ANS: Yes 20. ANS: No 2
14 21. ANS: even 22. ANS: neither 23. ANS: odd 24. ANS: even 25. ANS: even 26. ANS: 3
15 27. ANS: 28. ANS: y = 29. ANS: y = 30. ANS: y = ANS: 32. ANS: {1, 2, 3, 4, 5, 8, 9} 4
16 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: ANS: 38. ANS: ANS: 40. ANS: 41. ANS: 42. ANS: Not a polynomial 5
17 43. ANS: Polynomial; degree ANS: 3y 3 + 5y 2 + 5y ANS: 4x ANS: x 220x ANS: 3x 75x 3 ; remainder ANS: 6x  7; remainder ANS: x 2 + 5; remainder ANS: (3x  1)(3x + 1) 51. ANS: (3y  1)(9y 2 + 3y + 1) 52. ANS: (x + 1) ANS: (9x  7) ANS: 2(x + 2)(x  3) 6
18 55. ANS: (2x + 3)(5x + 3) 56. ANS: x 43x 3 + 9x 226x + 78; remainder ANS: 3x 2 + 3x  2; remainder ANS: Yes 59. ANS: No 60. ANS: 61. ANS: 4x ANS: ANS: ANS: ANS:
19 66. ANS: ANS: 68. ANS: No, no 69. ANS: No, no 70. ANS: f(x) = ; g(x) = x ANS: f(x) = ; g(x) = 5x ANS: {x 73. ANS: {x 74. ANS: Onetoone 75. ANS: No 8
20 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
21 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 ANS: Yes; degree ANS: f(x) = x 3 + 3x 24x  12 for a = ANS: f(x) = x 3 + x 26x for a = ANS: 7, multiplicity 1, crosses xaxis; 5, multiplicity 4, touches xaxis 91. ANS: 2, multiplicity 1, crosses xaxis; 4, multiplicity 3, crosses xaxis 92. ANS: xintercepts: 0, 2; yintercept: 0 10
22 93. ANS: xintercepts: 2, 5, 5; yintercept: 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
23 99. ANS: 100. ANS: x = 3, x = ANS: x =, x = ANS: none 103. ANS: none 12
24 104. ANS: 105. ANS: 106. ANS: (, 7) 107. ANS: (, 2) (2, 7) 13
25 ESSAY 108. ANS: 14
26 109. ANS: 15
11 /2 12 /2 13 /6 14 /14 15 /8 16 /8 17 /25 18 /2 19 /4 20 /8
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