College Algebra MATH 107 Spring, 2017, V4.8 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]; Range [-4, 2] Domain [-2, 1]; Range [0, 2] Domain (-, 2]; Range (-, 2] Domain [-4, 2]; Range [-2, 2] 2. Solve: A. B. C. D. 3x 5 = x 3 2. ______ No solution 2 7 2, 7 Page 1 of 11 College Algebra MATH 107 3. Determine the interval(s) on which the function is decreasing. A. (-, -1) B. (-, - 3) and (1, ) C. (- 2, 2) D. Spring, 2017, V4.8 3. ______ (- 4.5, - 1) and (2.5, ) 4. Determine whether the graph of y = 7 x is symmetric with respect to the origin, 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: | 3 - 8x | 21. A. B. C. D. 5. ______ [-9/4, 3] (-, 9/4] [3, ) (-, -9/4] [-9/4, ) Page 2 of 11 College Algebra MATH 107 Spring, 2017, V4.8 6. Which of the following represents the graph of 4x + 9y = 36 ? A. B. C. D. 6. ______ Page 3 of 11 College Algebra MATH 107 Spring, 2017, V4.8 7. Write a slope-intercept equation for a line parallel to the line x - 8y = 3 which passes through the point (16, -5). 7. ______ A. y = 8 x + 123 B. y= 1 x5 8 C. y= 1 x7 8 D. 1 y = x 3 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 a one-to-one relation. 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 an equivalent expression: 3 log (x + 4) - 2 log y + log 1 A. log ( 3x + 13 2 y ) B. x3 + 64 log 2 y C. 12 log( x) log(2 y ) D. ( x + 4 )3 log y2 10. Which of the functions corresponds to the graph? Spring, 2017, V4.8 9. ______ 10. ______ A. f ( x ) = e x 1 B. f ( x ) = 1 e x C. f ( x ) = 2 e x D. f ( x ) = e x Page 5 of 11 College Algebra MATH 107 Spring, 2017, V4.8 11. Suppose that for a function f , the equation f (x) = 0 has no real-number solutions. Which of the following statements MUST be true? A. B. C. D. 11. ______ f is an invertible function. f (x) > 0 for all x in the domain of f. f has no x-intercepts. f has no y-intercepts. 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) + 2 g(x) = f (x - 2) + 1 g(x) = f (x + 1) + 2 g(x) = f (x + 2) + 1 Page 6 of 11 College Algebra MATH 107 Spring, 2017, V4.8 SHORT ANSWER: 13. Multiply and simplify: (8 + 5i)(3 6i). Write the answer in the form a + bi, where a and b are real numbers. Answer: ________ 14. Solve, and write the answer in interval notation: x+4 0. x9 Answer: ________ 15. Water initially at 190 F. is left in a room of temperature 60 F to cool. - 0.096 t After t minutes, the temperature T of the water is given by T(t) = 60 + 130e 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 . 16 17. Solve: 57 x3 = 25 . Answer: ________ Answer: ________ 18. Suppose $4,200 is invested in an account at an annual interest rate of 6.6% 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 + 12x + 38. (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 Spring, 2017, V4.8 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) = Spring, 2017, V4.8 3x + 1 . x2 (a) State the domain. Answer: _________________ (b) State the vertical asymptote(s). Answer: _________________ (c) State the horizontal asymptote. Answer: _________________ (d) Which of the following represents the graph of f ( x) = 3x + 1 ? x2 GRAPH A. GRAPH B. GRAPH C. GRAPH D. Answer: ______________ Page 9 of 11 College Algebra MATH 107 Spring, 2017, V4.8 SHORT ANSWER, with work required to be shown, as indicated. 22. Let f (x) = x - 3 and g ( x) = x + 5 . f (a) Find (1) . Show work. g (b) Find the domain of the quotient function f . Explain. g 23. Points (-8, 9) and (-2, 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 x-axis. 24. Find the equation for a line which passes through the points (2, -1) and (7, -6). Write the equation in slope-intercept form. Show work. 25. Mary, a resident of Metropolis, pays Metropolis an annual tax of $80 plus 1.7% of her annual income. If Mary paid $1,015 in tax, what was Mary's income? Show work. 26. Let f (x) = 4x2 + 9 and g(x) = x - 2. (a) Find the composite function ( f o g )( x) and simplify. Show work. (b) Find ( f o g ) (1) . Show work. 2 27. Find the exact solutions and simplify as much as possible: 11x + 2 = 10x. Show work. 28. Given the function f ( x) = 5 1 x , find a formula for the inverse function. Show work. 8 8 Page 10 of 11 College Algebra MATH 107 Spring, 2017, V4.8 29. The fuel efficiency F for a particular car is given by F(x) = -0.018 x2 + 1.656 x + 4.4 where x is the speed of the car in miles per hour (mph) and F(x) is the corresponding fuel efficiency in miles per gallon (mpg). (a) What is the fuel efficiency if the car's speed is 70 mph? (b) What speed will yield the maximum fuel efficiency? Show work. 30. Solve: x+7 12 + 2 = 0 . Show work. x + 6 x 36 Page 11 of 11 Math 107 Final Examination Spring, 2017 1 Math 107College Algebra Name______________________________ Final Examination: Spring, 2017 Instructor __________________________ Answer Sheet Instructions: 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. Record your answers and work in this document. There are 30 problems. Problems #1-12 are multiple choice. Record your choice for each problem. Problems #13-21 are short answer. Record your answer for each problem. Problems #22-30 are short answer with work required. When requested, show all work and write all answers in the spaces allotted on the following pages. You may type your work using plaintext formatting or an equation editor, or you may hand-write your work and scan it. In either case, show work neatly and correctly, following standard mathematical conventions. Each step should follow clearly and completely from the previous step. If necessary, you may attach extra pages. You must complete the exam individually. Neither collaboration nor consultation with others is allowed. Your exam will receive a zero grade unless you complete the following honor statement. Please sign (or type) your name below the following honor statement: I have completed this final examinationmyself, working independently and not consulting anyone except the instructor. I have neither given nor received help on this final examination. Name _____________________ Date___________________ Math 107 Final Examination Spring, 2017 2 MULTIPLE CHOICE. Record your answer choices. 1. D 7. C 2. B 8. D 3. A 9. A 4. C 10. D 5. B 11. C 6. B 12. A SHORT ANSWER. Record your answers below. 13. 32-9i 14. [-6, 1) 15. 56.4 oF 16. -4 17. 7/8 18. 18.7 years 19. (a) (b) [-8, ) (6, -8) (c) (6, ) 20. (a) C (b) -2, 1, 3, 4 (c) (0, -6) (d) A 21. (a) (b) y=2 (c) x=5 Math 107 Final Examination (d) B Spring, 2017 3 Math 107 Final Examination Spring, 2017 4 SHORT ANSWER with Work Shown. Record your answers and work. Problem Number Solution Answers: (a)-2/3 (b) Work/for part (a) and explanation for part (b): a) f(x) = and g(x) = x-8 22 b) For domain, the numerator must be greater than or equal to 1, otherwise we will have negative in square root function which will produce imaginary value And the denominator must not be zero, which means we must not have 8 as x value otherwise it will make function value infinity So domain is 23 Answers: (a) (b)(-1, 2) (c)(-1, -2) Work for part (a): Length = Math 107 Final Examination Spring, 2017 5 Where (x1, y1) = (-6, 1) and (x2, y2) = (4, 3) Length = Length = Length = 2 Answer: y = -4x+5 Work: Here (x1, y1) = (3, -7) and (x2, y2) = (-1, 9) 24 Slope of line = So, we have slope = m = -4 Using the general equation of line (y-y1) = m(x-x1) We get y-(-7) = -4(x-3) y+7 = -4x + 12 y = -4x + 12- 7 y = -4x+5 Math 107 Final Examination Spring, 2017 6 Answer: $38,200 25 26 Work: Let x and y be the amount of sales and monthly salary for the person respectively. According to question, we have the following equation y = (6.5/100)x + 2100 we have to find the value of x when y = 4583 4583 = 0.065x + 2100 0.065x = 4583-2100 = 2483 x = 2483/0.065 = 38200 so, required amount of sales is $38,200 Answers: (a) 5x2 + 30x + 54 (b) 14 Work for part (a) and for part (b): (a) we have f(x) = 5x2+9 and g(x) = x + 3 (f o g)(x) = f[g(x)] = f[x+3] f(x+3) = 5(x+3)2+9 (f o g)(x) = 5(x2+6x+9)+9 (f o g)(x) = 5x2 + 30x + 54 (b) Here we have to set x = -4 in the (f o g)(x) (f o g)(-4) = 5(-4)2 + 30(-4) + 54 (f o g)(-4) = 5*16 -120 + 54 (f o g)(-4) = 80 -120 + 54 (f o g)(-4) = 134-120 (f o g)(-4) = 14 Math 107 Final Examination Spring, 2017 7 27 Answer: x = Work: We have 5x2-1= 8x Adding -8x on both sides, it gives 5x2-1-8x= 0 Writing it in standard form 5x2-8x-1 = 0 We can use the quadratic formula to find the roots of this equation When comparing it with the standard quadratic equation we get a= 5, b = -8 and c = -1 we know the formula for quadratic equation Setting the given values in the above formula, it gives This gives us two real values for x, i.e. Math 107 Final Examination Spring, 2017 8 Answer: 8x + 32 Work: The given function is f(x) = For inverse, let y = Adding 4 on both sides, we get y + 4 = (1/8)x multiplying both sides by 8, it gives 8y + 32 = x Replacing x and y We get y = f-1(x) = 8x + 32 28 29 Answers: (a) $510 (b) 1175 Work for (b): Number of donuts to maximize the profit is equal to the vertex point Math 107 Final Examination Spring, 2017 9 Number of donuts = -b/2a here b = 4.7 and a = -0.002 Number of donuts = -4.7/(2*(-0.002)) = 1175 So, required number of donuts to maximize the profit is 1175 donuts 30 Answer: x = -3 Work: Given expression is We can write x2-25 as (x-5)(x+5) The LCM of (x-5) and (x2-25) is (x2-25) which is equal to (x-5)(x+5) Multiplying both sides by LCM We get Cancelling out the common terms, it gives Solving it further Solving it using splitting the middle term, we get (x-5)(x+3) = 0 x = 5 and -3 We can only use x = -3 because x = 5 is an extraneous solution which makes the expression undefined, so the required value of x is -3 Math 107 Final Examination Spring, 2017 10