Use a graphing utility to approximate the real solutions, if any, of the equation rounded to two decimal places. 4) x3-6x + 3 = 0 (-5,5) 4)

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1 Advanced College Prep Pre-Calculus Midyear Exam Review Name Date Per List the intercepts for the graph of the equation. 1) x2 + y - 81 = 0 1) Graph the equation by plotting points. 2) y = -x ) List the intercepts of the graph. 3) 3) Use a graphing utility to approximate the real solutions, if any, of the equation rounded to two decimal places. 4) x3-6x + 3 = 0 (-5,5) 4) 5) -x4 + 3x x 2 = 9 x + 2 (-4,4) 5) 2 1

2 Graph the function. 6) y = 1 x 6) Determine whether the relation represents a function. If it is a function, state the domain and range. 7) Bob carrots Ann peas Dave squash 7) 8) {(1, -4), (-3, -3), (-3, 0), (6, 3), (22, 5)} 8) Determine whether the equation is a function. 9) y2 + x = 3 9) Find the value for the function. 10) Find f(7) when f(x) = x2 + 6x. 10) 11) Find f(-x) when f(x) = -3x2-4x ) 12) Find f(x - 1) when f(x) = 3x2 + 2x ) 13) Find f(x + h) when f(x) = 7x + 2 9x ) Find the domain of the function. 14) f(x) = 25 - x 14) 15) h(x) = x - 2 x3-64x 15) 2

3 For the given functions f and g, find the requested function and state its domain. 16) f(x) = 6x3-1; g(x) = 2x2 + 1 Find f g. 16) 17) f(x) = 9x + 8; g(x) = 2x + 6 Find f g. 17) 18) f(x) = 4x + 5 5x - 7 ; g(x) = 6x 5x - 7 Find f + g. 18) 19) f(x) = 4x - 5; g(x) = 2x - 4 Find f - g. 19) 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. 20) 20) The graph of a function f is given. Use the graph to answer the question. 21) Is f(60) positive or negative? 21)

4 22) How often does the line y = -10 intersect the graph? 22) Answer the question about the given function. 23) Given the function f(x) = x2 + 2x - 120, list the x-intercepts, if any, of the graph of f. 23) 24) Given the function f(x) = x 2-5, is the point (2, - 9) on the graph of f? 24) x ) Given the function f(x) = x 2 + 7, list the x-intercepts, if any, of the graph of f. 25) x - 2 The graph of a function is given. Determine whether the function is increasing, decreasing, or constant on the given interval. 26) (0, 1) 26) 4

5 27) (-1, 0) 27) 28) (- 3, 0) 28) 2 Use the graph to find the intervals on which it is increasing, decreasing, or constant. 29) 29) 5

6 The graph of a function f is given. Use the graph to answer the question. 30) Find the numbers, if any, at which f has a local maximum. What are the local maxima? 30) 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. 31) f(x) = x3-3x + 1, (-2, 2) 31) Solve the problem. 32) The cost C, in dollars, to produce graphing calculators is given by the function C(x) = 53x , where x is the number of calculators produced. How many calculators can be produced if the cost is limited to $129,700? 32) Find the vertex and axis of symmetry of the graph of the function. 33) f(x) = -x2-4x ) Determine the domain and the range of the function. 34) f(x) = -x2-6x ) Graph the function. 35) f(x) = x + 1 if x < 1-3 if x 1 35) 6

7 The graph of a piecewise-defined function is given. Write a definition for the function. 36) 36) Locate any intercepts of the function. 37) 1 if -6 x < -3 f(x) = x if -3 x < 6 3 x if 6 x 33 37) Based on the graph, find the range of y = f(x). 38) 1 x if x 0 f(x) = 2 8 if x = 0 38) 7

8 Graph the function by starting with the graph of the basic function and then using the techniques of shifting, compressing, stretching, and/or reflecting. 39) f(x) = (x + 3) ) 40) f(x) = (x - 3) ) 41) f(x) = x ) Find the function. 42) Find the function that is finally graphed after the following transformations are applied to the graph of y = x. The graph is reflected across the x-axis, stretched by a factor of 3, shifted right 3 units, and finally shifted vertically down 2 units. 42) 8

9 Find the function that is finally graphed after the following transformations are applied to the graph of y = x. 43) i) Reflect about the y-axis 43) ii) Shift down 7 units iii) Shift left 5 units Graph the function by starting with the graph of the basic function and then using the techniques of shifting, compressing, stretching, and/or reflecting. 44) f(x) = 1 6 x 2 44) 45) f(x) = 2 x 45) 9

10 46) f(x) = (-x)2 46) 47) f(x) = 3(x + 1) ) State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. 48) f(x) = 8 - x 2 48) 2 49) f(x) = x 5-8 x4 49) 50) f(x) = x(x - 11) 50) 51) f(x) = 3(x - 1)11(x + 1)4 51) Form a polynomial whose zeros and degree are given. 52) Zeros: -3, -5, 5; degree 3 52) 53) Zeros: 2, multiplicity 2; -2, multiplicity 2; degree 4 53) 10

11 For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. 54) f(x) = 1 5 x 4(x2-3) 54) 55) f(x) = 3(x2 + 2)(x - 3)2 55) Find the x- and y-intercepts of f. 56) f(x) = -x2(x + 2)(x2-1) 56) Determine the maximum number of turning points of f. 57) f(x) = (x - 2)2(x + 5)2 57) Analyze the graph of the given function f as follows: (a) Determine the end behavior: find the power function that the graph of f resembles for large values of x. (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) Graph f using a graphing utility. (e) Use the graph to determine the local maxima and local minima, if any exist. Round turning points to two decimal places. (f) Use the information obtained in (a) - (e) to draw a complete graph of f by hand. Label all intercepts and turning points. (g) Find the domain of f. Use the graph to find the range of f. (h) Use the graph to determine where f is increasing and where f is decreasing. 58) f(x) = x2(x2-4)(x + 4) 58) 59) f(x) = -x2(x - 1)(x + 3) 59) 60) f(x) = -2(x - 3)(x + 1)3 60) Find the domain of the rational function. 61) h(x) = x + 8 x2 + 9x 61) 62) h(x) = x + 2 x ) 63) f(x) = 2x2-4 3x2 + 6x ) 11

12 Use the graph to determine the domain and range of the function. 64) 64) Find the vertical asymptotes of the rational function. 65) h(x) = x + 8 x ) Give the equation of the horizontal asymptote, if any, of the function. 66) h(x) = 9x 2-5x - 9 2x2-6x ) 67) h(x) = 6x 3-3x - 8 8x ) x(x - 1) 68) f(x) = x 3 + 9x 68) Find the indicated intercept(s) of the graph of the function. x ) x-intercepts of f(x) = x2 + 5x ) 70) x-intercepts of f(x) = x 2 + 3x x2 + 3x ) 71) x-intercepts of f(x) = x 2 - x - 30 x ) 7x 72) y-intercept of f(x) = x ) 73) y-intercept of f(x) = (5x - 20)(x - 2) x2 + 8x ) 12

13 Graph the function. 74) f(x) = 2x - 4 x ) 75) f(x) = 4x (x - 4)(x + 4) 75) 76) f(x) = x 2 + 5x + 4 (x - 2)2 76) 13

14 Find the vertical asymptotes of the rational function. x ) f(x) = 16x - x3 77) For the given functions f and g, find the requested composite function value. 78) f(x) = 2x + 6; g(x) = 2x2 + 3; Find (f g)(2). 78) For the functions f and g and the number c, compute (f g)(c). 79) f(x) = x2 + 2x - 5 g(x) = x2-2x - 2 c = 5 79) Find the indicated composite for the pair of functions. 80) (f g)(x): f(x) = 6 x + 1, g(x) = 8 3x 80) 81) (f g)(x): f(x) = x + 9, g(x) = 8x ) Find the domain of the composite function f g. 82) f(x) = x + 7; g(x) = 7 x ) 83) f(x) = 6x + 18; g(x) = x 83) Indicate whether the function is one-to-one. 84) {(-8, -7), (-7, -7), (-6, -3), (-5, 1)} 84) Use the horizontal line test to determine whether the function is one-to-one. 85) 85) Find the inverse. Determine whether the inverse represents a function. 86) {(6, -7), (-2, -6), (-4, -5), (-6, -4)} 86) 14

15 The graph of a one-to-one function f is given. Draw the graph of the inverse function f-1 as a dashed line or curve. 87) f(x) = x ) The function f is one-to-one. Find its inverse. 3 88) f(x) = 7x ) Find the inverse function of f. State the domain and range of f. 89) f(x) = 3x - 2 x ) Decide whether or not the functions are inverses of each other. 90) f(x) = 8-9x; g(x) = x (x - 8) 90) 9 Use the graph of the given one-to-one function to sketch the graph of the inverse function. For convenience, the graph of y = x is also given. 91) 91) Solve the problem. 92) Show that f and g are inverse functions or state that they are not. f(x) = 3-8x - 6 ; g(x) = - x ) 15

16 Approximate the value using a calculator. Express answer rounded to three decimal places. 93) 3 93) 94) e ) 95) e1.2 95) 96) 2e 96) Solve the equation. 97) x = 8 97) 98) 4 -x = ) 99) 3x = 27 99) 100) 4(3x - 5 ) = ) 101) 44x - 4 = 83x 101) Change the exponential expression to an equivalent expression involving a logarithm. 102) 6 3 = ) 103) 3 2 = x 103) 104) 11 x = ) 105) ex = ) Change the logarithmic expression to an equivalent expression involving an exponent. 106) log 1/2 16 = ) 107) log b 32 = 5 107) 108) ln z = 7 108) Find the exact value of the logarithmic expression. 109) log ) 110) log ) 16

17 111) ln e 111) Use a calculator to find the natural logarithm correct to four decimal places. 112) ln ) Solve the equation. 113) log 8 x2 = 4 113) 114) log 8 (x2-7x) = 1 114) 115) ln x = 6 115) Use the properties of logarithms to find the exact value of the expression. Do not use a calculator. 116) log ) 117) ln e2 117) 118) log log ) 119) log log ) 120) log ) 121) 5 log ) Write as the sum and/or difference of logarithms. Express powers as factors. 122) log ) 123) log 15 4 x y 123) 124) log 5 x2 y7 124) 125) log 5 x - 6 x7 125) 126) log n2m 126) 17

18 127) log 5 8 m 9 n k2 127) 128) ln (x + 4)(x - 6) (x - 9)2 2/3 128) Express as a single logarithm. 129) log c m + log c n 129) 130) 2 log b q - log b r 130) 131) 2 log c m log c n log c j - 5 log c k 131) 132) 3 loga (2x + 1) - 2 loga (2x - 1) ) 133) ln x 2 + 3x - 18 x ln x 2 + 4x - 12 x ln (x2-6x + 9) 133) Use the Change-of-Base Formula and a calculator to evaluate the logarithm. Round your answer to three decimal places. 134) log ) 135) log ) Use the Change-of-Base Formula and a calculator to evaluate the logarithm. Round your answer to two decimal places. 136) log ) 137) log ) 138) log (2/3) ) Solve the equation. 139) log y 11 = 3 139) 140) log 2 (x + 2) = 3 140) 141) log (x + 4) = log (4x - 2) 141) 142) log (4 + x) - log (x - 5) = log 2 142) 143) log 4 (x + 5) + log 4 (x - 1) = 2 143) 18

19 Solve the exponential equation. Express the solution set in terms of natural logarithms. 144) 8 4x = ) 145) 5 x + 8 = 6 145) Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 1 x 146) = ) 4 147) 2(x - 1) = ) 19