MAT 107 College Algebra Fall 2013 Name. Final Exam, Version X

Transcription

1 MAT 107 College Algebra Fall 013 Name Final Exam, Version X EKU ID Instructor Part 1: No calculators are allowed on this section. Show all work on your paper. Circle your answer. Each question is worth 10 points. 1. f(x) = x + 10; g(x) = 9. Find the composite function (f g)(x) and its domain. x + 3. Solve: x - 3 x + 8 > 0 3. Let f(x) = x3 - x - 11x is a root of f(x). Find all roots of f(x), including any complex roots.

2 4. Express log 3 (x + 8) - log 3 (x + 1) as the logarithm of a single expression. Assume that variables represent positive numbers. 5. Solve for t: 5t - 7t = 1 6. Solve: (7 + 3x) = 1 4 of 8 (Part 1 is pages 1 - ; Part is pages 3-8)

3 1. Find f(x - 1) when f(x) = 5x - x - 1. A) 5x - 1x + B) 5x - 1x + 6 C) 5x - 7x + D) -1x + 5x + 6. The graph of a function f is given. How often does the line y = 5 intersect the graph F) once G) twice H) three times J) zero times 3. For the function f(x) = x 3 - x, find f(x) - f(1) x - 1 when x 1 A) x + x B) x - x C) 1 D) x3 - x - 1 x Solve: (x + 7) = 3. Express the solution in terms of natural logarithms. F) x = ln + 7 G) x = ln 3 - ln - ln 7 ln 3 H) x = ln ln 3 + ln 7 J) x = ln 3 ln A rocket is launched from the top of a cliff that is 11 feet high with an initial velocity of 11 feet per second. The height, h(t), of the rocket after t seconds is given by the equation h(t) = -16t + 11t How long after the rocket is launced will it strike the ground? Round to the nearest tenth of a second, if necessary. A) 8.7 sec B) 9.3 sec C) 8.3 sec D) 7.9 sec 3 of 8 (Part 1 is pages 1 - ; Part is pages 3-8)

4 6. Which of the following is the graph of this function? x + 1 if -7 x < -4 f(x) = -5 if x = -4 -x + 4 if x > -4 F) G) H) J) 7. Find the slope of the line. A) 15 B) - 15 C) - 15 D) 15 4 of 8 (Part 1 is pages 1 - ; Part is pages 3-8)

5 8. The number of books in a small library increases according to the function B = 8800e0.0t, where t is measured in years. How many books will the library have after 5 years? F) 0,63 books G) 976 books H) 11,079 books J) 8800 books 9. Sales for a small clothing company can be modeled by the linear function S(x) = 4437x + 73,677, where x is the number of years since 005 and S(x) is in dollars. Find the sales in 011. A) $446,499 B) $95,86 C) $100,99 D) $468, Solve the inequality (x + 1) and put the answer in interval notation. F) [-7, 5] G) (-9, 3) H) (-7, 5) J) [-9, 3] 11. Find the function that is finally graphed after the following transformations are applied to the graph of y = x. The graph is shifted up 7 units, reflected about the y-axis, and finally shifted left 4 units. A) y = -x B) y = -x C) y = - x D) y = -x Find the intervals on which the function is increasing, decreasing, or constant. F) Increasing on (-3, -) and (, 4); decreasing on (-1, 1); constant on (-, -1) and (1, ) G) Decreasing on (-3, -1) and (1, 4); increasing on (-, 1) H) Decreasing on (-3, -) and (, 4); increasing on (-1, 1) J) Decreasing on (-3, -) and (, 4); increasing on (-1, 1); constant on (-, -1) and (1, ) 5 of 8 (Part 1 is pages 1 - ; Part is pages 3-8)

6 13. Find the domain of the rational function f(x) = x - 4 3x + 6x - 7 A) {x x -6, 4} B) {x x -4, 6} C) {x x -6, -4, 4} D) all real numbers 14. Use the Leading Coefficient Test to determine the end behavior of the polynomial function f(x) = -5 x 3 + M x + 4 x + 4 for some real number M. F) rises to the left and rises to the right G) rises to the left and falls to the right H) falls to the left and rises to the right J) falls to the left and falls to the right 15. Find the zeros of the quadratic function F(x) = x + 4x - 16 A) x =, x = - 4 B) x =, x = 4 C) x = -, x = - 4 D) x =-, x = The graph of a function f is given. Find the numbers, if any, at which f has a local minimum. What are the local maxima? F) f has a local minimum at x = -3.3 and -.5; the local minimum at -3.3 is -.5; the local minimum at -.5 is 5 G) f has a local minimum at x = -.5 and 5; the local minimum at -.5 is -3.3; the local minimum at 5 is -.5 H) f has a local maximum at x = -3.3 and -.5; the local maximum at -3.3 is -.5; the local maximum at -.5 is 5 J) f has a local maximum at x = -.5 and 5; the local maximum at -.5 is -3.3; the local maximum at 5 is of 8 (Part 1 is pages 1 - ; Part is pages 3-8)

7 17. f(x) = 7x + 9, g(x) = /x; Find (g f)(3). A) 41 B) C) 9 3 D) Find the x- and y-intercepts of f(x) = 8x - x 3. F) x-intercepts: 0,, - ; y-intercept: 8 G) x-intercepts: 0,, - ; y-intercept: 0 H) x-intercepts: 0, -8; y-intercept: 8 J) x-intercepts: 0, -8; y-intercept: Use the graph of the rational function to find the equation of the horizontal asymptote. A) x = -1 B) y = -1 C) y = 1 D) x = 1 0. Find the vertex and axis of symmetry of the graph of the function f(x) = 3x - 6x F) (-1, -3); x = -1 G) (1, -3); x = 1 H) (1, 0); x = 1 J) (-1, 0); x = Find the vertical asymptotes, if any, of the graph of the rational function. g(x) = A) x = B) x =, x = -, x = 0 C) x =, x = - D) no vertical asymptote x x - 4. Solve the inequality x 3 + 3x - 8x > 0 and put the answer in interval notation. F) (-4, 0) or (7, ) G) (-7, ) H) (-,-7) or (0, 4) J) (-7, 0) or (4, ) 7 of 8 (Part 1 is pages 1 - ; Part is pages 3-8)

8 3. List the potential rational zeros of the polynomial function f(x) = x 5 + kx3-4x + 3x + 5, where k is an integer. Do not find the zeros. A) ± 1 4, ± 5 4, ± 5 B) ± 5, ± 1 5 C) ± 1, ± 1 5 D) ± 1, ± 5 4. Form a polynomial f(x) with real coefficients, degree: 4 and zeros: i and -5i F) f(x) = x 4 + 9x G) f(x) = x 4 - x 3 + 9x H) f(x) = x 4 + 9x -5x J) f(x) = x 4-5x The function f(x) = (x + ) 3-8 is one-to-one. Find its inverse. A) f -1 (x) = 3 x B) f -1 (x) = 3 x + 10 C) f -1 (x) = 3 x D) f -1 (x) = 3 x Change the exponential expression 4 3/ = 8 to an equivalent expression involving a logarithm. F) log 8 log 3 4 = 4 G) log 8 4 = 3 H) log 3 4 = 3 J) log 4 8 = 3 7. Use properties of logarithms to expand the logarithmic expression log 5 x - 4 x 8. A) 8log 5 x -log 5 (x - 4) B) log 5 (x - 4) + 8log 5 x C) log 5 (x - 4) -log 5 x D) log 5 (x - 4) - 8log 5 x 8. Solve: log 6 (b - ) = 1 F) b = 4 G) b = -1 H) b = 8 J) b = 3 8 of 8 (Part 1 is pages 1 - ; Part is pages 3-8)