Class: Math 1650.500 (Pre-Calc) Test 1 Review Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. Solve the inequality. Express the solution using interval notation. 1 4 x + 1 < 4x 9 9 a. b. c. d. e. ____ 1 x( , ) 7 x ( , 7) 1 x ( , ) 5 x ( , 5) 1 x( , ) 5 2. Solve the inequality. Express the solution using interval notation. 1 17 3x 29 a. b. c. d. e. [ -4, 6 ] [ -6, 4 ] [ -4, 6 ) [ -6, 4 ) ( -4, 6 ) 1 Name: ________________________ ____ ID: A 3. Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. x2 + x > 6 ____ a. ÊÁ , 2 ˆ˜ ÊÁ 3, ˆ˜ ; Ë ¯ Ë ¯ b. ÊÁ , 3 ˆ˜ ÊÁ 1, ˆ˜ ; Ë ¯ Ë ¯ c. ÊÁ , 5 ˆ˜ ÊÁ 2, ˆ˜ ; Ë ¯ Ë ¯ d. ÊÁ , 3 ˆ˜ ÊÁ 2, ˆ˜ ; Ë ¯ Ë ¯ e. ÊÁ , 2 ˆ˜ ÊÁ 2, ˆ˜ ; Ë ¯ Ë ¯ 4. Solve the inequality for x, assuming that m, q, and c are positive constants. m( qx c ) qc a. x b. x c. x d. x e. x q( m + c ) mc c( q m) mq c( m + q ) mq m( m q ) cq m( c + q ) cq 2 Name: ________________________ ____ 5. Determine the correct equation for the line with an x-intercept of –8 and a y-intercept of 16. a. b. c. d. e. ____ b. c. d. e. y = 5x 2 1 y= x2 5 1 y= x+2 5 1 y= x5 5 y = 5x + 2 7. Write the equation for the line passing through the point (7, 5) which is perpendicular to the line y = –15. a. b. c. d. e. ____ y = 2x 16 y = 2x + 16 1 y = x + 16 2 1 y = x 16 2 y = 2x + 16 6. Determine the correct equation for the line passing through the point (30, –8) and parallel to the line x + 5y = 20. a. ____ ID: A x=6 x=7 y=8 y=9 y = 7 8. Determine the correct equation for the line passing through the point (3, 3) which is parallel to the line passing through both of the points (5, 4) and (–7, 112). a. b. c. d. e. y = 9x 30 1 y = 9x + 30 y = 9x + 30 1 y = 9x 30 1 y = 9x + 30 3 Name: ________________________ ____ 9. Find the slope and the y-intercept of the line 11 ,y intercept = 22 12 a. m= b. m = 1 ,y intercept = 0 c. 5m = ID: A 1 1 x y + 2 = 0 and draw its graph. 11 12 11 ,y intercept = 22 12 d. m= e. m = 1,y intercept = 11 11 ,y intercept = 24 12 4 Name: ________________________ ID: A ____ 10. Express the following rule in function notation: "square, add 5, then take the square root" x + 5) 2 a. f (x) = ( b. f (x) = ( x + 5) 2 c. f (x) = (x + 4) 2 d. f (x) = x +5 e. f (x) = x2 + 5 ____ 11. Evaluate the function f (x) a. b. c. d. e. 19 x at f (3). 7x f (3) = 1.5 f (3) = 1.2 f (3) = 1.6 f (3) = 1.3 f (3) = 1.4 ____ 12. Evaluate the following piecewise defined function at f (1), f (5) and f (4). ÔÏÔ Ô 2 f (x) = ÔÌÔ x ÔÔ ÔÓ 3x a. b. c. d. e. + 2x if x 4 if x > 4 f (1) = 3, f (5) = 15, f (4) = 29 f (1) = 3, f (5) = 15, f (4) = 24 f (1) = 3, f (5) = 19, f (4) = 24 f (1) = 3, f (5) = 15, f (4) = 22 f (1) = 3, f (5) = 19, f (4) = 29 5 Name: ________________________ ID: A ____ 13. For the function f (x) = 3x 2 + 3, find f (a + h) f (a) , h 0. h a. b. c. d. e. f (a + h) f (a) h f (a + h) f (a) h f (a + h) f (a) h f (a + h) f (a) h f (a + h) f (a) h = 3h + 6a = 3h + 3a = 9h + 6a = 6h + 3a = 3h + 12a ____ 14. Find the domain of the following function: f (x) = 9 x 4 a. b. c. d. e. ( , ) [4, ) [0, ) (0, ) ( , 4] ____ 15. Find the domain of the following function: f (x) = a. b. c. d. e. x x 11x + 10 2 [0, 1) (1, 10) (10, ) [1, 10] ( , 1) (1, 10) (10, ) (1, 10) ( , ) 6 Name: ________________________ ID: A ____ 16. Select the correct graph and domain of the function. f (x) = 4x + 1 a. d. 1 Domain: [ ,) 4 b. 1 Domain: [ ,) 4 e. Domain: [0,) Domain: [0,) c. 1 Domain: [ ,6) 4 7 Name: ________________________ ID: A ____ 17. Sketch the graph of the piecewise defined function. ÏÔ Ô 0 if x 2 f(x) ÔÌÔ ÔÔÓ 1 if x 2 a. d. b. e. c. 8 Name: ________________________ ID: A ____ 18. Sketch the graph of the piecewise defined function. ÏÔ ÔÔ 2 if x 2 ÔÔ f(x) ÌÔ x if 2 x 2 ÔÔ ÔÔ 2 if x 2 Ó a. d. b. e. c. 9 Name: ________________________ ID: A ____ 19. Determine whether the equation defines y as a function of x. x 2 3y 3 a. no b. yes ____ 20. Determine the interval on which the function in the graph below is decreasing. a. b. c. d. e. [4, –1] [–2, 7] [–3, 6] [–7, –3] [6, 10] 10 Name: ________________________ ID: A ____ 21. Determine the interval on which the function in the graph below is decreasing. a. b. c. d. e. [–2, 10] [4, 2] [–3, 9] [9, 13] [–7, –3] ____ 22. The graph of the function y = –x 2 + 8x is: Find its maximum or minimum value. a. min = 16 b. max = 24 c. min = –32 d. min = –16 e. max = 16 11 Name: ________________________ ID: A ____ 23. The graph of a function is given below. What is the average rate of change of the function between the indicated values of the variable? a. b. c. d. e. 1 6 1 6 1 6 ____ 24. What is the average rate of change of the function f (x) = 9x 8 between x = 2 and x = 3? a. b. c. d. e. 9 12 6 10 8 ____ 25. What is the average rate of change of the function f (t) = t 2 7t between t 5 and t 4? a. b. c. d. e. –16 –10 –14 –13 –8 12 Name: ________________________ ID: A ____ 26. Determine the average rate of change for the function f (x) = a. b. c. d. e. 1 3 1 27 1 9 1 27 1 3 ____ 27. What is the average rate of change of the function f (x) = a. b. c. d. e. 1 between x 1 and x 3. x x between x 9 and x 16? 1 49 1 7 1 7 1 49 7 ____ 28. The table below shows the number of CD players sold in a small electronics store in the years 1989-1999: Year 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 CD players sold 540 700 730 655 690 565 615 540 620 650 640 What was the average rate of change of sales between 1989 and 1999? a. b. c. d. e. 640 CD players/year 64 CD players/year 100 CD players/year 10 CD players/year 25 CD players/year 13 Name: ________________________ ID: A ____ 29. The table below shows the number of CD players sold in a small electronics store in the years 1989-1999. Year 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 CD players sold 525 525 540 650 650 650 725 725 790 795 595 Between which two successive years did CD player sales increase most quickly? a. b. c. d. e. 1991, 1992 1995, 1996 1989, 1990 1997, 1998 none of these ____ 30. A 100-m race ends in a three-way tie for first place. The graph shows distance as a function of time for each of the three winners. Find the average speed for each winner. a. b. c. d. e. A: 10 m/sec, B: 10 m/sec, C: 10 m/sec A: 5 m/sec, B: 5 m/sec, C: 10 m/sec A: 10 m/sec, B: 5 m/sec, C: 10 m/sec A: 5 m/sec, B: 5 m/sec, C: 5 m/sec A: 10 m/sec, B: 5 m/sec, C: 5 m/sec 14 Name: ________________________ ID: A ____ 31. Suppose the graph of f is given. Describe how the graph of the function can be obtained from the graph of f. y 4f (x 5) 3 a. Shift the graph of y f(x) to the left 5 units, stretch vertically by a factor of 4, and then shift upward 3 units. b. Shift the graph of y f(x) to the right 4 units, stretch vertically by a factor of 5, and then shift downward 3 units. c. Shift the graph of y f(x) to the right 5 units, stretch vertically by a factor of 4, and then shift downward 3 units. d. Shift the graph of y f(x) to the left 5 units, stretch vertically by a factor of 4, and then shift downward 3 units. e. Shift the graph of y f(x) to the left 4 units, stretch vertically by a factor of 5, and then shift downward 3 units. ____ 32. Explain how the graphs of g and h are obtained from the graph of f. f(x) = x 2 , g(x) = (x + 4) 2 , h(x) = x 2 + 4 a. b. c. d. e. The graph of g is obtained from the graph of f by moving up 4 units, and the graph of h is obtained from the graph of f by shifting left 4 units. The graph of g is obtained from the graph of f by shifting left 5 units, and the graph of h is obtained from the graph of f by moving up 5 units. The graph of g is obtained from the graph of f by shifting left 2 units, and the graph of h is obtained from the graph of f by moving up 2 units. The graph of g is obtained from the graph of f shifting right 4 units, and the graph of h is obtained from the graph of f moving up 4 units. The graph of g is obtained from the graph of f by shifting left 4 units, and the graph of h is obtained from the graph of f by moving up 4 units. 15 Name: ________________________ ID: A ____ 33. Explain how the graph of g is obtained from the graph of f. f(x) a. x , g(x) 1 2 x5 The graph of g(x) 1 2 x 5 is obtained by shifting the graph of f(x) units, and then shrinking the graph vertically by a factor of b. The graph of g(x) 1 2 1 . 2 x 5 is obtained by shifting the graph of f(x) 5 units, and then shrinking the graph vertically by a factor of c. The graph of g(x) x upward 5 x to the left 1 . 2 1 2 x 5 is obtained by shifting the graph of f(x) x to the right 1 2 x 5 is obtained by shifting the graph of f(x) x to the right 5 units. d. The graph of g(x) 5 units, and then shrinking the graph vertically by a factor of e. The graph of g(x) 1 2 x 5 is obtained by shifting the graph of f(x) 5 units. ____ 34. Find the domain of g û f, if f (x) = x 2 and g(x) = a. ( , 3] [ b. c. d. e. f. ( , 3) ( 3 ,) ( , 3) (3,) x3 x 3 ( , 3] [3,) 3 ,) ____ 35. For f (x) = x 5 + 5, g (x) = x 8, and h (x) = x. Find f û g û h. a. (f û g û h)(x) = ( x 8) 5 + 5 b. (f û g û h)(x) = ( x 3) 5 c. (f û g û h)(x) = x 5 + x 3 + d. (f û g û h)(x) = (x + 5) (x 8) e. (f û g û h)(x) = 5 1 . 2 x x x5 3 16 x3. x downward Name: ________________________ ID: A ____ 36. Express the function in the form f û g . x3 x3 2 x f(x) , g(x) x 3 x2 x f(x) x 3 , g(x) x2 x f(x) x 3 , g(x) x2 G(x) a. b. c. d. f(x) x 3 2, g(x) e. f(x) x x2 x , g(x) x 3 x2 ____ 37. Express the function in the form f û g . H(x) 4 x a. f(x) x 4 , g(x) b. f(x) x , g(x) c. f(x) 4 x , g(x) x 2 d. f(x) 4 x , g(x) e. f(x) x , g(x) x 4x x 4x 17 Name: ________________________ ID: A ____ 38. An airplane is flying at a speed of 250 mph at an altitude of 4 miles. The plane passes directly above a radar station at time t = 0. Find the distance s between the plane and the radar station after 4 minutes. a. b. c. d. e. 17.6 miles 17.1 miles 16.6 miles 18.1 miles 19.0 miles ____ 39. Find the inverse function of f (x) = 10x + 20. a. f b. f c. f d. f e. f 1 1 1 1 1 (x) = (x) = (x) = (x) = (x) = 20 x 10 1 10x+20 10 x 20 10 x 20 x 20 10 18 Name: ________________________ ____ 40. Find the inverse function of f (x) = a. f b. f c. f d. f e. f ID: A 1 . x+2 1 +2 x 1 (x) = x + 2 1 1 (x) = 2 x 1 (x) = x 2 1 1 (x) = x2 1 (x) = ____ 41. Find the inverse function of f (x) = 2 + a. b. f f c. f d. f e. f 1 (x) = 3 x. 1 2+ 3 x 1 (x) = x 2 1 (x) = (2 x) 3 1 (x) = (x 2) 3 1 (x) = (x + 2) 3 3 3 19 Name: ________________________ ID: A ____ 42. A function f is given. f(x) x5 Sketch the graph of f. Use the graph of f to sketch the graph of f 1 . Find f 1 . d. a. f 1 (x) f 1 x 2 5 x5 b. e. f 1 f 1 (x) x 5 x4 c. f 1 (x) x 2 5 20 Name: ________________________ ID: A ____ 43. The given function f (x) = (x + 4) 2 is not one-to-one. Find a restricted domain for which the function is one-to-one. Also find the inverse of the function with the restricted domain. a. b. c. d. e. x 4; f 1 x 4; f 1 x 4; f 1 x 7; f 1 x 7; f 1 (x) = (x) = x +4 x +4 (x) = x 4 (x) = x +4 (x) = x 4 ____ 44. Find the vertex of the given parabola. y = x 2 2x + 10 a. b. c. d. e. (1, 9) (2, 10) (1, 7) (1, 10) (1, 11) ____ 45. Find the vertex of the given parabola. y = 6x 2 + 12x 11 a. b. c. d. e. (1, 11) (1, 17) (1, 17) (12, 11) (6, 25) 21 Name: ________________________ ID: A ____ 46. Express the quadratic function in standard form and find its maximum or minimum value. f (x) = x 2 + 6x 7 a. f (x) = ( x + 3 ) 2 6; Maximum f ( 3) = 6 b. f (x) = ( x + 6 ) 2 17; Minimum f ( 6) = 17 c. f (x) = ( x + 3 ) 2 7; Maximum f (3) = 7 d. f (x) = ( x + 4 ) 2 16; Minimum f (4) = 16 e. f (x) = ( x + 3 ) 2 16; Minimum f ( 3) = 16 ____ 47. Find the minimum or maximum value of the function. f (x) = x 2 + x + 9 a. b. c. d. e. f ( 9) = 35 4 1 35 f( ) = 2 2 35 1 f( ) = 2 2 35 1 f( ) = 4 2 1 31 f( ) = 2 4 ____ 48. Find the maximum or minimum value of the function. f(x) = 9 8x 8x 2 a. b. c. d. e. 2 f(x) = ( x + 2 ) + 11; Maximum f(2) 11 2 ÁÊ ÁÊ 1 ˆ˜ 1 ˜ˆ f(x) ÁÁÁÁ x ˜˜˜˜ 13; Maximum f ÁÁÁÁ ˜˜˜˜ 13 2¯ Ë Ë 2¯ ÁÊ f(x) 8 ÁÁÁÁ x Ë ÊÁ f(x) = 8 ÁÁÁÁ x + Ë 2 1 ˜ˆ˜˜ 11; Maximum 2 ˜˜¯ 1 ˜ˆ˜˜ 2 ˜˜¯ 2 ÁÊ 1 ˆ˜ f ÁÁÁÁ ˜˜˜˜ 11 Ë 2¯ ÁÊ 1 ˆ˜ + 11; Minimum f ÁÁÁÁ ˜˜˜˜ = 11 Ë 2¯ 2 ÁÊ 1 ˜ˆ f(x) ÁÁÁÁ x ˜˜˜˜ 9; Minimum 2¯ Ë ÁÊ 1 ˆ˜ f ÁÁÁÁ ˜˜˜˜ 9 Ë 2¯ 22 Name: ________________________ ID: A ____ 49. If a ball is thrown directly upward with a velocity of 80 ft/s, its height (in feet) after t seconds is given by y = 80t 16t 2 . What is the maximum height attained by the ball? a. b. c. d. e. 25 feet 176 feet 50 feet 100 feet 80 feet ____ 50. A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function R (x) = 384x 0.8x 2 where the revenue R (x) is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum? a. b. c. d. e. $46,090, 230 units $46,080, 240 units $46,070, 250 units $46,080, 245 units $0, 480 units ____ 51. Determine the end behavior of the graph of the function. y = 8x 3 – 7x 2 + 3x + 7 a. y as x , and y as x b. y as x , and y as x c. y as x , and y as x d. y as x , and y as x e. y as x , and y as x ____ 52. How many local maxima and minima does the polynomial have? y = 9x 2 + 7x + 6 a. b. c. d. e. 1 maximum and 1 minimum 1 maximum and 2 minima 0 maxima and 1 minimum 1 maximum and 0 minima 2 maxima and 1 minimum 23 Name: ________________________ ID: A ____ 53. How many local maxima and minima does the polynomial have? y = 0.2x 5 + 2x 4 22.33x 3 12.5x 2 + 1,050x + 14 a. b. c. d. e. 2 maxima and 2 minima 1 maxima 5 minima 2 maxima 1 minima 0 maxima and 1 minima 1 maxima and 0 minima ____ 54. The point P is on the unit circle. The x-coordinate of P is a. b. c. d. ÊÁ ˆ˜ ÁÁ 1 ˜ ÁÁ , 48 ˜˜˜ ÁÁ 7 7 ˜˜˜ Á Ë ¯ ÁÊÁ 1 6 ˜ˆ˜ ÁÁ , ˜˜ Á7 7˜ Ë ¯ ÊÁ ˆ˜ ÁÁ 1 ˜ ÁÁ , 48 ˜˜˜ ÁÁ 7 7 ˜˜˜ Á Ë ¯ ÊÁ ˆ˜ 48 ˜˜˜ ÁÁÁ 1 ÁÁ , ˜ ÁÁ 7 7 ˜˜˜ Ë ¯ 1 and P is in quadrant I. Find the point P(x, y). 7 2 ____ 55. The point P is on the unit circle. The x-coordinate of P is and P is in quadrant II. Find the point P (x, y). 7 ˆ ÁÊÁ 2 ˜ 45 ˜˜˜ Á ˜ a. ÁÁÁ , ÁÁ 7 7 ˜˜˜ Ë ¯ ÊÁ ˆ ÁÁ 2 45 ˜˜˜˜ Á ˜ b. ÁÁ , ÁÁ 7 7 ˜˜˜ Ë ¯ ÊÁ 2 5 ˆ˜ c. ÁÁÁÁ , ˜˜˜˜ Ë 7 7¯ ˆ ÊÁ ÁÁ 2 45 ˜˜˜˜ Á ˜ d. ÁÁ , ÁÁ 7 7 ˜˜˜ ¯ Ë ____ 56. Find the terminal point P (x, y) on the unit circle determined by t hundredth. a. (0.93, 0.73) b. (0.81, –0.59) c. (–0.81, –0.59) d. (0.81, 0.59) 24 5 . Round the values to the nearest Name: ________________________ ____ 57. Find the reference number for t a. b. c. d. ID: A 5 . 4 5 5 4 4 4 2 . Round the coordinates of the 3 terminal point to the nearest hundredth, and express the reference number in terms of ____ 58. Find the terminal point P (x, y) and the reference number determined by t a. terminal point is (–0.50, 0.87), reference number is b. terminal point is (–0.50, –0.87), reference number is c. terminal point is (0.50, –0.87), reference number is d. terminal point is (0.50, 0.87), reference number is ____ 59. Find the exact value of the trigonometric functions sin a. b. c. 3 3 3 2 3 13 11 and sin . 2 2 13 11 1, sin 0 2 2 13 11 sin 1, sin 1 2 2 13 11 sin 0, sin 1 2 2 sin 9 9 ____ 60. Find the exact values of the trigonometric functions sin and cos . 4 4 a. 3 3 9 9 sin , cos 2 2 4 4 b. 3 9 1 9 sin , cos 2 4 2 4 c. 2 2 9 9 sin , cos 2 2 4 4 25 Name: ________________________ ID: A 7 1 ____ 61. Find the exact values of the trigonometric functions sin and sin . 6 6 7 1 a. sin 0.5, sin 0.5 6 6 7 1 b. sin 0.5, sin 0.5 6 6 7 1 c. sin 0.5, sin 0.5 6 6 ____ 62. Find the exact value of the trigonometric function cos a. b. c. 15 . 2 15 1 2 15 cos 0 2 15 cos 1 2 cos ÊÁ ˆ ÁÁ 3 7 ˜˜˜˜ Á ˜ . Find sin t, cos t, and tan t. ____ 63. The terminal point determined by t is ÁÁ , ÁÁ 4 4 ˜˜˜ Ë ¯ 7 3 7 a. sin t , cost , tan t 4 4 3 b. sint 3 7 7 , cost , tant 4 3 4 c. sint 7 7 3 , cost , tant 4 3 4 ____ 64. Find the values of the trigonometric functions of t if sint a. sint 5 5 2 5 2 , cost , tan t , cot t 3 2 5 3 b. sin t 5 2 5 2 5 , cost , tan t , cot t 3 5 3 2 c. sint 5 5 2 5 5 , cost , tan t , cot t 3 2 5 3 5 and the terminal point of t is in quadrant II. 3 ____ 65. Determine whether the function |5x| cos 2x is even, odd, or neither. a. neither b. odd c. even 26 Test 1 Review Answer Section MULTIPLE CHOICE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: C A D C B B B C A E C B A A A A E E B C C E A A A E C D A D D E D A A E D B E C 1 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: D C C E B E D C D B B D A A A D D A B C C B C A C 2
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