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Math 135 Final Exam Study Guide The graph of a function is given. Follow the directive(s). 1) y 5 (0. 5, 2) (3. 5, 2) 5 (6, -1. 1) x -5 (-5, -3) (-4, -3) -5 (a) List all the intervals on which the function is increasing. (b) List all the intervals on which the function is decreasing. (c) List all the intervals on which the function is constant. (d) Find the domain. (e) Find the range. (f) Find f(-5). (g) Find f(6). (h) Find x when f(x) = 0. (i) Find the x-intercept(s).

(j) Find the y-intercept(s). Find the value for the function. 2) Find f(3) when f(x) = x2 – 3x + 6. ) Find f(-2) when f(x) = x2 – 9 . x+3 4) Find f(-9) when f(x) = |x|- 6. Determine whether the relation represents a function. If it is a function, state the domain and range. 5) Bob Ann Dave carrots peas squash 1 6) Bob Ann Dave Ms. Lee Mr. Bar Find the domain of the function. 3x 7) g(x) = x2 – 1 8) f(x) = x2 x2 + 1 4-x 9) f(x) = For the given functions f and g, find the requested function and state its domain.

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10) f(x) = 7 – 6x; g(x) = -9x + 6 Find f + g. 11) f(x) = 6 – 2x; g(x) = -4x + 2 Find f + g. 12) f(x) = 4x – 2; g(x) = 8x – 3 Find f – g. 13) f(x) = 8x – 6; g(x) = 2x – 5 Find f – g. 4) f(x) = 5x + 1; g(x) = 3x – 1 Find f ? g. 15) f(x) = 5x + 3; g(x) = 4x + 8 Find f ? g. 16) f(x) = 5x + 4; g(x) = 4x – 3 f Find . g 17) f(x) = 5x + 3; g(x) = 4x – 5 f Find . g 2 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. 18) 5 y -5 5 x -5 19) 10 y 5 -10 -5 5 x -5 -10 Graph the function. 20) f(x) = -x + 3 2x – 3 if x &lt; 2 if x ? 2 y 5 -5 5 x -5 3 21) f(x) = x+5 -8 -x + 5 if -8 ? x &lt; -2 if x = -2 if x &gt; -2 y 10 5 -10 -5 -5 -10 5 10 x

Find the indicated composite for the pair of functions. 2) (f † g)(x): f(x) = 6x + 9, g(x) = 5x – 1 23) (f † g)(x): f(x) = 3x + 10, g(x) = 3x – 1 24) (g † f)(x): f(x) = 4×2 + 2x + 7, g(x) = 2x – 8 25) (g † f)(x): f(x) = 4×2 + 2x + 6, g(x) = 2x – 7 Follow the directive(s). 26) Find and simplify the difference quotient of f, f(x + h) – f(x) , h ? 0, for the given function: f(x) = x 2 – 7x – 2 h 27) Find and simplify the difference quotient of f, f(x + h) – f(x) , h ? 0, for the given function: f(x) = x 2 + 6x + 4 h 4 28) f(x) = -x2 – 2x + 8 y 10 5 -10 -5 -5 5 10 x -10 For the given function above, (a) Find the vertex. b) Find the x- and y- intercepts. (c) Find the axis of symmetry. (d) Graph the function. 29) f(x) = x2 – 2x – 3 y 10 5 -10 -5 -5 -10 5 10 x For the given function above, (a) Find the vertex. (b) Find the x- and y- intercepts. (c) Find the axis of symmetry. (d) Graph the function. 5 The graph of a function f is given. Use the horizontal line test to determine whether f is one-to-one. 30) y 10 -10 10 x -10 31) y 10 -10 10 x -10 If the following defines a one-to-one function, find the inverse. 32) f(x) = 5x + 2 33) f(x) = x2 – 2 34) f(x) = 2 x-2 Solve the equation. Find all real solutions. + 2x 35) 2 = 32 7 – 3x 1 36) 4 = 16 37) log9 (x – 2) + log9 (x – 2) = 1 38) log (x – 3) = 1 – log x Find the domain of the function. 39) f(x) = log (x – 10) 6 40) f(x) = log (x – 8) Solve the problem. 41) A rumor is spread at an elementary school with 1200 students according to the model N = 1200(1 – e-0. 16d) where N is the number of students who have heard the rumor and d is the number of days that have elapsed since the rumor began. How many students will have heard the rumor after 5 days? 42) If \$5,000 is invested for 6 years at 5%, compounded continuously, find the future value.

Compute the amount in m years if a principal P is invested at a nominal annual interest rate of r compounded as given. Round to the nearest cent. 43) P = \$1,000, m = 8, r = 11% compounded annually 44) P = \$1,000, m = 9, r = 12% compounded semiannually 45) P = \$480, m = 5, r = 11% compounded quarterly Solve the system of equations by substitution. 46) x + 5y = 5 7x – 2y = -2 47) x 2y + = 32 3 2 x 5y = 40 9 4 Solve the system of equations. 48) 2x + 7y = -6 2x + 2y = 24 49) x + y = -9 x – y = 18 50) 1 2 26 x+ y= 2 5 5 5x + 2y = 56 51) 4x – 9y = 2 8x – 18y = 10 52) 7x + y = 7 -14x – 2y = -14 7

Give the equation of the specified asymptote(s). x-4 53) Vertical asymptote(s): f(x) = x2 – 4 54) Vertical asymptote(s): f(x) = x – x3 x2 – 4x + 3 3x + 10 x2 + 12x + 35 55) Vertical asymptote(s): f(x) = State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. 56) f(x) = 2x + 7×5 57) f(x) = -9×4 – 5×3 + 3 58) f(x) = 1 + 13 x 59) f(x) = x3/2 – x6 – 8 60) f(x) = 5 – x2 7 2 1 – x 3 3 61) f(x) = 62) f(x) = x(x – 13) Solve the equation. Find all real solutions. 63) x3 + 2×2 – 5x – 6 = 0 64) 2×3 – x2 – 10x + 5 = 0 Form a polynomial whose zeros and degree are given. 5) Zeros: 0, – 7, 6; degree 3 66) Zeros: -3, -2, 3; degree 3 67) Zeros: -1, 1, – 5; degree 3 List the potential rational zeros of the polynomial function. Do not find the zeros. 68) f(x) = 5×3 – x2 + 3 69) f(x) = 6×4 + 3×3 – 4×2 + 2 70) f(x) = x5 – 6×2 + 6x + 7 Use synthetic division to find the rmaining zeros of the polynomial. 71) f(x) = x3 – 3×2 – 5x + 39; zero: -3 8 72) f(x) = x3 – 2×2 – 11x + 52; zero: -4 Find all of the real zeros of the polynomial function, then use the real zeros to factor f over the real numbers. 73) f(x) = x3 + 2×2 – 9x – 18 Find all the (real and complex) zeros of the polynomial function. 4) f(x) = x4 – 8×3 + 16×2 + 8x – 17 9 Answer Key Testname: 135STUDYGUIDE 1) decreasing (a) (-4, 0. 5); (b) (3. 5, 6); (c) (-5, -4) U (0. 5, 3. 5); (d) (-5, 6); (e) (-3, 2); (f) -3; (g) -1. 1; (h) -1. 2; 5. 2; (i) -1. 2; 5. 2; (j) 1. 4 2) 6 3) – 5 4) 3 5) not a function 6) function domain: {Bob, Ann, Dave} range: {Ms. Lee, Mr. Bar} 7) {x|x ? -1, 1} 8) all real numbers 9) {x|x ? 4} 10) (f + g)(x) = -15x + 13; all real numbers 11) (f + g)(x) = -6x + 8; all real numbers 12) (f – g)(x) = -4x + 1;

all real numbers 13) (f – g)(x) = 6x – 1; all real numbers 14) (f ? )(x) = 15×2 – 2x – 1; all real numbers 15) (f ? g)(x) = 20×2 + 52x + 24; all real numbers 5x + 4 3 f ; {x|x ? } 16) ( )(x) = 4x – 3 4 g 5x + 3 5 f ; {x|x ? } 17) ( )(x) = 4x – 5 4 g 18) function domain: {x|x ? -2} range: {y|y ? 0} x- intercepts: (-2, 0), (2, 0) y-intercept: (0, -2) 19) not a function 20) y 5 -5 5 x -5 10 Answer Key Testname: 135STUDYGUIDE 21) y 10 (-2, 7) (-2, 3) 5 -10 (-8, -3) -5 -5 (-2, -8) -10 5 10 x 22) 30x + 3 23) 9x + 7 24) 8×2 + 4x + 6 25) 26) 27) 28) 8×2 + 4x + 5 2x + h – 7 2x + h + 6 vertex (-1, 9) x-intercepts (2, 0), (- 4, 0) y-intercept (0, 8) Axis of symmetry: x = -1 y 10 -10 -5 -5 -10 5 10 x 29) vertex (1, -4) intercepts (3, 0), (- 1, 0), (0, -3) y 10 5 -10 -5 -5 -10 5 10 x 11 Answer Key Testname: 135STUDYGUIDE 30) Yes 31) No x-2 32) f-1(x) = 5 33) Not a one-to-one function 2x + 2 34) f-1(x) = x 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 49) 50) 51) 52) 53) 54) 55) 56) 57) 58) 59) 60) 61) 62) 63) 2 3 5 5 x &gt; 10 x&gt;8 661 students \$6749 \$2304. 54 \$2854. 34 \$825. 81 x = 0, y = 1 x = 100, y = -27 x = 18, y = -6 x = 4. 5, y = -13. x = 12, y = -2 inconsistent y = -7x + 7, where x is any real number x = 2, x = -2 x=3 x = -7, x = -5 Yes; degree 5 Yes; degree 4 No; x is raised to a negative power No; x is raised to non-integer 3/2 power Yes; degree 2 Yes; degree 1 Yes; degree 2 {-3, -1, 2} 1 64) { , 5, – 5 } 2 65) f(x) = x3 + x2 – 42x for a = 1 66) f(x) = x3 + 2×2 – 9x – 18 for a = 1 67) f(x) = x3 + 5×2 – x – 5 for a = 1 3 1 68) ± , ± , ± 1, ± 3 5 5 1 1 2 1 69) ± , ± , ± , ± , ± 1, ± 2 3 2 3 6 70) ± 1, ± 7 71) 3 + 2i, 3 – 2i 72) 3 + 2i, 3 – 2i 12 Answer Key Testname: 135STUDYGUIDE 73) -3, -2, 3; f(x) = (x + 3)(x + 2)(x – 3) 74) 1, -1, 4 – i, 4 + i 13

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