258 CHAPTER 4 Polynomial and Rational Functions remove this discontinuity by defining the rational function R using the following piecewise-defined function: ( ) = − + − ≠ = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ ⎪ R x x x x x x 2 5 2 4 if 2 3 4 if 2 2 2 (a) Redefine R from Problem 33 so that the discontinuity at = x 3 is removed. (b) Redefine R from Problem 35 so that the discontinuity at = x 3 2 is removed. 68. Challenge Problem Removing a Discontinuity Refer to Problem 67. (a) Redefine R from Problem 34 so that the discontinuity at = − x 5 is removed. (b) Redefine R from Problem 36 so that the discontinuity at = − x 5 2 is removed. 66. Texting Speed A study of a new keyboard layout for smartphones found that the average number of words users could text per minute could be approximated by ( ) ( ) = + + N t t t 32 2 5 where t is the number of days of practice with the keyboard. (a) What was the average number of words users could text with the new layout at the beginning of the study? (b) What was the average number of words users could text after using the layout for 1 week? (c) Find and interpret the horizontal asymptote of N. 67. Challenge Problem Removing a Discontinuity In Example 5, we graphed the rational function ( ) = − + − R x x x x 2 5 2 4 2 2 and found that the graph has a hole at the point ( ) 2, 3 4 . Therefore, the graph of R is discontinuous at ( ) 2, 3 4 . We can Explaining Concepts 69. Graph each of the following functions: = − − = − − = − − = − − y x x y x x y x x y x x 1 1 1 1 1 1 1 1 2 3 4 5 Is = x 1 a vertical asymptote? Why? What happens for = x 1? What do you conjecture about the graph of = − − ≥ y x x n 1 1 , 1 n an integer, for = x 1? 70. Graph each of the following functions: = − = − = − = − y x x y x x y x x y x x 1 1 1 1 2 4 6 8 What similarities do you see? What differences? 71. Create a rational function that has the following characteristics: crosses the x-axis at 3; touches the x-axis at −2; one vertical asymptote, = x 1; and one horizontal asymptote, = y 2. Give your rational function to a fellow classmate and ask for a written critique of your rational function. 72. Create a rational function that has the following characteristics: crosses the x-axis at 2; touches the x-axis at −1; one vertical asymptote at = − x 5 and another at = x 6; and one horizontal asymptote, = y 3. Compare your function to a fellow classmate’s. How do they differ? What are their similarities? 73. Write a few paragraphs that provide a general strategy for graphing a rational function. Be sure to mention the following: proper, improper, intercepts, and asymptotes. 74. Create a rational function with the following characteristics: three real zeros, one of multiplicity 2; y-intercept 1; vertical asymptotes, = − x 2 and = x 3; oblique asymptote, = + y x2 1. Is this rational function unique? Compare your function with those of other students. What will be the same as everyone else’s? Add some more characteristics, such as symmetry or naming the real zeros. How does this modify the rational function? 7 5. Explain the circumstances under which the graph of a rational function has a hole. Retain Your Knowledge Problems 76–85 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. 76. Subtract: ( ) ( ) − + − − + x x x x 4 7 1 5 9 3 3 2 77. Solve: + = − + x x x x 3 3 1 2 5 78. Find the absolute maximum of ( ) = − + − f x x x 2 3 6 5. 2 79. Simplify: ( ) ( ) ( ) ( ) ( ) − − − − − − − x x x x x x 1 2 2 3 2 2 1 2 2 2 2 80. Find the function whose graph is the same as the graph of = y x but shifted down 4 units. 81. Find ( ) g 3 where ( ) = − < − ≥ ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪⎪ g x x x x x x 3 7 if 0 5 9 if 0 2 82. Given ( ) = + − f x x x3 2, 2 find ( ) − f x 2 . 83. Determine whether the lines = − y x3 2 and + = x y 2 6 7 are parallel, perpendicular, or neither. 84. Solve: − + = x x 7 5 85. Solve: − − + − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = x x x 2 4 4 0 2 2 2 ‘Are You Prepared?’ Answers 1. ( ) ( ) ( ) − 0, 1 4 , 1,0 , 1,0
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