College Algebra MATH 107 Summer, 2017, V1.9 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. 1. ______ 4 2 -4 -2 2 4 -2 -4 A. B. C. D. Domain [–2, 2]; Domain [–2, 0]; Domain [–3, 1]; Domain [–1, 1]; 2. Solve: A. B. C. D. 3x − 5 = x − 3 Range [–3, 1] Range [0, 2] Range [–2, 2] Range [0, 2] 2. ______ No solution 2 2, 7 7 Page 1 of 11 College Algebra MATH 107 3. Determine the interval(s) on which the function is decreasing. A. ( −∞, 1) B. (– 0.5, 3) C. (1, 5) D. (3, 6.5) Summer, 2017, V1.9 3. ______ and ( 5, ∞ ) 4. Determine whether the graph of y = 5 − x is symmetric with respect to the origin, the x-axis, or the y-axis. 4. ______ A. not symmetric with respect to the x-axis, not symmetric with respect to the y-axis, and not symmetric with respect to the origin B. symmetric with respect to the x-axis only C. symmetric with respect to the y-axis only D. symmetric with respect to the origin only 5. Solve, and express the answer in interval notation: | 9 – 7x | ≤ 5. A. B. C. D. 5. ______ (–∞, 2] ∪ [4/7, ∞) (–∞, 4/7] ∪ [2, ∞) [4/7, 2] [4/7, ∞) Page 2 of 11 College Algebra MATH 107 Summer, 2017, V1.9 6. Which of the following represents the graph of 9x − 2y = 18 ? A. B. C. D. 6. ______ Page 3 of 11 College Algebra MATH 107 Summer, 2017, V1.9 7. Write a slope-intercept equation for a line parallel to the line x + 8y = 3 which passes through the point (–24, 7). 7. ______ A. y = −8 x − 185 B. 1 y =− x+4 8 C. 1 y = − x+7 8 D. y= 1 x + 10 8 8. Which of the following best describes the graph? A. B. C. D. 8. ______ It is the graph of a one-to-one function. It is the graph of a function but not one-to-one. It is not the graph of a function. It is a parabola. Page 4 of 11 College Algebra MATH 107 9. Express as an equivalent expression: 4 log x – log (y + 6) + log 1 A. log ( 4 x − y − 5) B. log(4 x) log( y + 6) C.  4x +1 log    y+6  D.  x4  log    y+6 10. Which of the functions corresponds to the graph? Summer, 2017, V1.9 9. ______ 10. ______ A. f ( x ) = e x − 1 B. f ( x ) = −e x C. f ( x ) = e− x + 1 D. f ( x ) = −e x + 1 Page 5 of 11 College Algebra MATH 107 Summer, 2017, V1.9 11. Suppose that for a function f , the equation f (x) = 0 has exactly three real-number solutions. Which of the following statements MUST be true? A. B. C. D. f f f f 11. ______ has exactly three y-intercepts. has exactly three x-intercepts. is a polynomial of degree 3. 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)? 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 – 1) + 1 g(x) = f (x – 1) – 1 g(x) = f (x + 1) – 1 g(x) = f (x + 1) + 1 Page 6 of 11 College Algebra MATH 107 Summer, 2017, V1.9 SHORT ANSWER: 13. Multiply and simplify: (2 + 5i)(3 + 7i). 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+4 ≤ 0. x −1 Answer: ________ Answer: ________ 15. Water initially at 185° F. is left in a room of temperature 70° F to cool. – 0.096 t After t minutes, the temperature T of the water is given by T(t) = 70 + 115e Find the temperature of the water 20 minutes after it is left to cool. (Round to the nearest degree.) Answer: ________  1  16. Find the value of the logarithm: log 4   .  64  17. Solve: 75x−4 = 49 . Answer: ________ Answer: ________ 18. Suppose $8,300 is invested in an account at an annual interest rate of 5.7% compounded continuously. How long (to the nearest tenth of a year) will it take the investment to double in size? Answer: ________ 2 19. Let f (x) = x + 6x + 10. (a) Find the vertex. Answer: ________ (b) State the range of the function. Answer: ________ (c) On what interval is the function increasing? Answer: ________ Page 7 of 11 College Algebra MATH 107 Summer, 2017, V1.9 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 Answer: ________ (b) State the y-intercept. Answer: ________________ (c) State the zeros of the function. Answer: ________________ 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) = Summer, 2017, V1.9 2+ x . 3− x (a) State the domain. Answer: _________________ (b) State the horizontal asymptote. Answer: _________________ (c) State the vertical asymptote(s). Answer: _________________ (d) Which of the following represents the graph of f ( x) = 2+ x ? 3− x GRAPH A. GRAPH B. GRAPH C. GRAPH D. Answer: ______________ Page 9 of 11 College Algebra MATH 107 Summer, 2017, V1.9 SHORT ANSWER, with work required to be shown, as indicated. 22. Let f ( x) = x + 5 and f (x) = x – 3 . f  (a) Find  (−1) . Show work. g (b) Find the domain of the quotient function f . Explain. g 23. Points (2, 1) and (6, –3) are endpoints of the diameter of a circle. (a) What is the length of the diameter? Give the exact answer, simplified as much as possible. Show work. (b) What is the center point C of the circle? (c) Given the point C you found in part (b), state the point symmetric to C about the y-axis. 24. Find the equation for a line which passes through the points (6, 2) and (9, –7). Write the equation in slope-intercept form. Show work. 25. A salesperson earns a base salary of $1,475 per month and a commission of 8.4% on the amount of sales. If the salesperson has a paycheck of $4,637.60 for one month, what was the amount of sales for the month? Show work. 26. Let f (x) = 5x2 + 7 and g(x) = x – 3. (a) Find the composite function ( f o g )( x) and simplify. Show work. (b) Find ( f o g ) (−2) . Show work. 2 27. Find the exact solutions and simplify as much as possible: 11x + 2 = 10x. Show work. 1 28. Given the function f ( x) = 3 + x , find a formula for the inverse function. Show work. 8 Page 10 of 11 College Algebra MATH 107 Summer, 2017, V1.9 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 + 25.50x – 180 (a) What is the company’s profit if 40 tourists are given the tour? (b) How many tourists should be given the tour in order to maximize the company’s profit? Show work. 30. Solve: x−7 20 + 2 = 0 . Show work. x − 5 x − 25 Page 11 of 11
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