86 CHAPTER 2 Functions and Their Graphs 50. The graph below represents the speed v (in miles per hour) of Michael’s car as a function of time t (in minutes). v(t ) (2, 30) (4, 30) (4.2, 0) (6, 0) (9.1, 0) (7.4, 50) (8, 38) (7.6, 38) (7, 50) t (a) Over what interval of time was Michael traveling fastest? (b) Over what interval(s) of time was Michael’s speed zero? (c) What was Michael’s speed between 0 and 2 minutes? (d) What was Michael’s speed between 4.2 and 6 minutes? (e) What was Michael’s speed between 7 and 7.4 minutes? (f) When was Michael’s speed constant? 51. Graph a function whose domain is x x x 3 8, 5 { } − ≤ ≤ ≠ and whose range is y y y 1 2, 0 { } − ≤ ≤ ≠ What point(s) in the rectangle x y 3 8, 1 2 − ≤ ≤ − ≤ ≤ cannot be on the graph? Compare your graph with those of other students. What differences do you see? 52. Is there a function whose graph is symmetric with respect to the x-axis? Explain. 53. Explain why the vertical-line test works. 47. Consider the following scenario: Barbara decides to take a walk. She leaves home, walks 2 blocks in 5 minutes at a constant speed, and realizes that she forgot to lock the door. So Barbara runs home in 1 minute. While at her doorstep, it takes her 1 minute to find her keys and lock the door. Barbara walks 5 blocks in 15 minutes and then decides to jog home. It takes her 7 minutes to get home. Draw a graph of Barbara’s distance from home (in blocks) as a function of time. 48. Consider the following scenario: Jayne enjoys riding her bicycle through the woods. At the forest preserve, she gets on her bicycle and rides up a 2000-foot incline in 10 minutes. She then travels down the incline in 3 minutes. The next 5000 feet is level terrain, and she covers the distance in 20 minutes. She rests for 15 minutes. Jayne then travels 10,000 feet in 30 minutes. Draw a graph of Jayne’s distance traveled (in feet) as a function of time. 49. The graph below represents the distance d (in miles) that Sobia was from home as a function of time t (in hours). Answer the questions by referring to the graph. In parts (a)–(g), how many hours elapsed and how far was Sobia from home during the times listed? d(t ) (2, 3) (2.5, 3) (2.8, 0) (3, 0) (4.2, 2.8) (5.3, 0) (3.9, 2.8) t (a) From t 0 = to t 2 = (b) From t 2 = to t 2.5 = (c) From t 2.5 = to t 2.8 = (d) From t 2.8 = to t 3 = (e) From t 3 = to t 3.9 = (f) From t 3.9 = to t 4.2 = (g) From t 4.2 = to t 5.3 = (h) What is the farthest distance that Sobia was from home? (i) How many times did Sobia return home? ‘Are You Prepared?’ Answers 1. 4,0, 4,0, 0, 2, 0,2 ( ) ( ) ( ) ( ) − − 2. False Retain Your Knowledge Problems 54–63 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 54. If f x x x 3, 2 ( ) = − + − find f x 2 . ( ) − 55. Find the distance between the points 3, 6 ( ) − and 1,0 . ( ) 56. Write an equation of the line with slope 2 3 that contains the point 6, 4 . ( ) − 57. Find the domain of g x x 4 5. 3 ( ) = + − 58. Find the number that must be added to m m 12 2 − to complete the square. 59. Rationalize the numerator: x x 6 6 − − 60. Two cars leave an intersection at the same time, one traveling north at 25 mph and the other traveling west at 35 mph. How long will it take for the cars to be 40 miles apart? 61. Find the numbers x that satisfy both of the inequalities x x 3 4 7and5 2 13 + ≤ − < 62. Simplify x x x 5 7 2 8 10 . 2 ( ) ( ) − + − − 63. Write the inequality x 3 10 − ≤ ≤ in interval notation.
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