College Algebra MATH 107 Spring, 2017, V4.11 MATH 107 FINAL EXAMINATION This is an open-book exam. You may refer to your text and other course materials as you work on the exam, and you may use a calculator. You must complete the exam individually. Neither collaboration nor consultation with others is allowed. Record your answers and work on the separate answer sheet provided. There are 30 problems. Problems #1-12 are Multiple Choice. Problems #13-21 are Short Answer. (Work not required to be shown) Problems #22-30 are Short Answer with work required to be shown. MULTIPLE CHOICE 1. Determine the domain and range of the piecewise function. A 1. ______ 4 2 -4 2 -2 4 -2 -4 A. B. C. D. Domain [-2, 2]; Range [-3, 1] Domain [-2, -1/2]; Range [0, 1] Domain [-1, 1]; Range [-3, 1] Domain [-3, 1]; Range [-2, 2] 2. Solve: 10 3x = x A. B. C. D. A 2. ______ No solution -5, 2 -5 5 Page 1 of 11 College Algebra MATH 107 Spring, 2017, V4.11 3. Determine the interval(s) on which the function is increasing. A. ( 1.25, 3.25) B. ( , C. ( 1/ 3, 3) D. ( ,1) and ( 4, ) C 3. ______ 1/ 3) and (3, 4.75) and ( 4.75, ) 4. Determine whether the graph of y = 2 x is symmetric with respect to the origin, B the x-axis, or the y-axis. 4. ______ A. symmetric with respect to the x-axis only B. symmetric with respect to the y-axis only C. symmetric with respect to the origin only D. not symmetric with respect to the x-axis, not symmetric with respect to the y-axis, and not symmetric with respect to the origin 5. Solve, and express the answer in interval notation: A. B. C. D. 7 3x 17 . A 5. ______ (-, -10/3] (-, -10/3] [8, ) [-10/3, 8] [-8, 10/3] Page 2 of 11 College Algebra MATH 107 Spring, 2017, V4.11 6. Which of the following represents the graph of 7x 2y = 14 ? A. B. C. D. A 6. ______ Page 3 of 11 College Algebra MATH 107 Spring, 2017, V4.11 7. Write a slope-intercept equation for a line parallel to the line x - 8y = 16 which passes through B the point (24, -1). 7. ______ A. 1 y = x2 8 B. y= C. y = 2 x + 47 D. 1 y = x 1 8 1 x4 8 8. Which of the following best describes the graph? A. B. C. D. B 8. ______ It is the graph of a function and it is one-to-one. It is the graph of a function but not one-to-one. It is the graph of an absolute value relation. It is not the graph of a function. Page 4 of 11 College Algebra MATH 107 9. Express as a single logarithm: Spring, 2017, V4.11 log x - 3 log y + log 1 A 9. ______ 3 A. x +1 log y B. x +1 log 3 y C. x log 3 y D. log ( x + 1 3 y ) 10. Which of the functions corresponds to the graph? B 10. ______ A. f ( x ) = e x + 4 B. f ( x ) = e x + 4 C. f ( x ) = e x + 3 D. f ( x ) = e x + 3 Page 5 of 11 College Algebra MATH 107 Spring, 2017, V4.11 11. Suppose that for a function f , the equation f (x) = 0 has exactly three real-number solutions. B 11. ______ Which of the following statements MUST be true? A. B. C. D. f f f f is a polynomial of degree 3. has exactly three x-intercepts. has exactly three y-intercepts. is an invertible function. 12. The graph of y = f (x) is shown at the left and the graph of y = g(x) is shown at the right. (No formulas are given.) What is the relationship between g(x) and f (x)? C 12. ______ -4 4 4 2 2 2 -2 4 -4 2 -2 -2 -2 -4 -4 y = f (x) 4 y = g(x) A. B. C. D. g(x) = f (x - 2) - 5 g(x) = f (x - 5) - 2 g(x) = f (x + 2) - 5 g(x) = f (x) - 5 Page 6 of 11 College Algebra MATH 107 Spring, 2017, V4.11 SHORT ANSWER: 13. Multiply and simplify: (7 - 2i)(5 + i). Write the answer in the form a + bi, where a and b are real numbers. 14. Solve, and write the answer in interval notation: x-8 => x-1 x-8 => 0 0< x-1 x =>8 or x< -1 x 8 0. x +1 33-3i Answer: ________ x< -1 or x => 8 Answer: ________ 15. A bowl of soup at 195 F. is placed in a room of constant temperature of 65 F. The temperature T of the soup t minutes after it is placed in the room is given by T(t) = 65 + 130 e - 0.075 t Find the temperature of the soup 28 minutes after it is placed in the room. (Round to the nearest tenth of a degree.) 81 Answer: ________ 1 16. Find the value of the logarithm: log 3 . 9 17. Solve: 745 x = 49 . 1/729 Answer: ________ 0 Answer: ________ 18. Suppose $5,500 is invested in an account at an annual interest rate of 7.3% compounded continuously. How long (to the nearest tenth of a year) will it take the investment to double in 10 size? Answer: ________ 2 19. Let f (x) = x 10x + 18. (a) Find the vertex. ( 5, -7) Answer: ________ (b) State the range of the function. [-7, infinity) Answer: ________ (c) On what interval is the function decreasing? (infinity, 5) Answer: ________ Page 7 of 11 College Algebra MATH 107 Spring, 2017, V4.11 20. Consider the polynomial P(x), shown in both standard form and factored form. 1 1 17 1 P ( x) = x 3 + x 2 + x + 3 = ( x + 3)( x + 1)( x 5) 5 5 5 5 (a) Which sketch illustrates the end behavior of the polynomial function? A. B. vvvv C. vvvv D. vvvv vvvv b Answer: ________ (b) State the zeros of the function. ( -3,0), ( -1,0), (5,0) Answer: ________________ (c) State the y-intercept. (0,3) Answer: ________________ a Answer: ________ (d) State which graph below is the graph of P(x). GRAPH A GRAPH B GRAPH C GRAPH D Page 8 of 11 College Algebra MATH 107 21. Let f ( x) = Spring, 2017, V4.11 2+ x . 3 x (a) State the domain. (3,0) Answer: _________________ (b) State the horizontal asymptote. y= -1 Answer: _________________ (c) State the vertical asymptote(s). x=3 Answer: _________________ (d) Which of the following represents the graph of f ( x) = 2+ x ? 3 x GRAPH A. GRAPH B. GRAPH C. GRAPH D. d Answer: ______________ Page 9 of 11 College Algebra MATH 107 Spring, 2017, V4.11 SHORT ANSWER, with work required to be shown, as indicated. 22. Let f ( x) = radical (-5+9)/-5-1 radical (4)/-6 2/-6 = - 1/3 x + 9 and g(x) = x 1. f (a) Find (5) . Show work. g (b) Find the domain of the quotient function f . Explain. g There are no varibles in the expression the domain is all real numbers. {xlx, = R} 23. Points (3, -7) and (5, -9) are endpoints of the diameter of a circle. 3+5/2, -7+-9/2 8/2, -16/2 (4,-8) is centerpoint (a) What is the length of the diameter? Give the exact answer, simplified as much as possible. d=radical (5-3)^2+(-9-(-7))^2 Show work. (b) What is the center point C of the circle? d=radical 4+4 d=2 radical 2 is diameter (c) Given the point C you found in part (b), state the point symmetric to C about the y-axis. (0,-8) 24. Find the equation for a line which passes through the points (-6, -5) and (-3, 1) . Write the equation in slope-intercept form. Show work. y-(-3)= 2(x-1) = y+3= 2x-2 = y=2x-2-3 == y=2x-1 1-(-5)/ -3-(-6) 1+5/-3+6 6/3 =2 25. Amy, a resident of Metropolis, pays Metropolis an annual tax of $32 plus 2.6% of her annual income. If Amy paid $2,307 in tax, what was Amy's income? Show work. 2307/,026= 88,730.77-32= 88,698.77 88,698.77+32 = 88,730.77 * .026= 2307 26. Let f (x) = 5x2 + 4 and g(x) = x + 2. f(x+2) 5(x+2)(x+2)+4 5(x^2+2x+2x+4)+4 5x^2+20x+20+4 =5x^2+20x+24 (a) Find the composite function ( f o g )( x) and simplify. Show work. (b) Find ( f o g ) (3) . Show work. f( -3+2) f(-1) = 5(-1)^2 + 4 5+4= 9 0= -8x^2+8x+3 2 27. Find the exact solutions and simplify as much as possible: 8x = 3 + 8x. Show work. 28. Given the function f ( x) = 1 x 4 , find a formula for the inverse function. Show work. 7 y= 7x + 28 y = 1/7x -4 x = 1/7y -4 4+x = 1/7y 28 +7x = y Page 10 of 11 College Algebra MATH 107 Spring, 2017, V4.11 29. The Travel Time bus company has determined that when x tourists are given a particular bus tour, the profit P, in dollars, is given by P(x) = -0.25 x2 + 27.50x - 240 P(38)= -.25(38)^2 + 27.5(38) - 240 P(38)=444 (a) What is the company's profit if 38 tourists are given the tour? $444 (b) How many tourists should be given the tour in order to maximize the company's profit? Show work. 30. Solve: x+5 28 = 2 . Show work. x + 7 x 49 x+5/x+7 = 28/x^2 -49 x+5/x+7 = 28/ (x+7)(x-7) (x+5)(x-7)/(x+7)(x-7) = 28/ (x+7)(x-7) x^2 -7x+5x-35=28 x^2-2x-63 =0 x^2-9x+7x-63=0 x(x-9)+7(x-9)=0 (x-9)=0 (x+7)=0 x=9 and x= -7 Page 11 of 11