310 CHAPTER 2 Graphs and Functions 90. Concept Check Suppose ƒ1x2 is an odd function and g1x2 is an even function. Fill in the missing entries in the table. x −2 −1 0 1 2 ƒ1x2 0 -2 g1x2 0 2 1 1ƒ∘ g21x2 1 -2 91. Show that 1ƒ∘ g21x2 is not equivalent to 1g∘ ƒ21x2 for ƒ1x2 = 3x - 2 and g1x2 = 2x - 3. 92. Show that for the functions ƒ1x2 = x3 + 7 and g1x2 = 23 x - 7, both 1ƒ∘ g21x2 and 1g∘ ƒ21x2 equal x. For certain pairs of functions ƒ and g, 1ƒ∘ g21x2 = x and 1g∘ ƒ21x2 = x. Show that this is true for each pair in the following exercises. 93. ƒ1x2 = 4x + 2, g1x2 = 1 41x - 22 94. ƒ1x2 = -3x, g1x2 = - 1 3 x 95. ƒ1x2 = 23 5x + 4, g1x2 = 1 5 x3 - 4 5 96. ƒ1x2 = 23 x + 1, g1x2 = x3 - 1 Find functions ƒ and g such that 1ƒ∘ g21x2 = h1x2. (There are many possible ways to do this.) See Example 9. 97. h1x2 = 16x - 222 98. h1x2 = 111x2 + 12x22 99. h1x2 = 2x2 - 1 100. h1x2 = 12x - 323 101. h1x2 = 26x + 12 102. h1x2 = 23 2x + 3 - 4 (Modeling) Solve each problem. 103. Relationship of Measurement Units The function ƒ1x2 = 12x computes the number of inches in x feet, and the function g1x2 = 5280x computes the number of feet in x miles. What is 1ƒ∘ g21x2, and what does it compute? 104. Relationship of Measurement Units The function ƒ1x2 = 3x computes the number of feet in x yards, and the function g1x2 = 1760x computes the number of yards in x miles. What is 1ƒ∘ g21x2, and what does it compute? 105. Area of an Equilateral Triangle The area of an equilateral triangle with sides of length x is given by the function 1x2 = 23 4 x 2. (a) Find 12x2, the function representing the area of an equilateral triangle with sides of length twice the original length. (b) Find the area of an equilateral triangle with side length 16. Use the formula 12x2 found in part (a). x x x 106. Perimeter of a Square The perimeter x of a square with side of length s is given by the formula x = 4s. (a) Solve for s in terms of x. (b) If y represents the area of this square, write y as a function of the perimeter x. (c) Use the composite function of part (b) to find the area of a square with perimeter 6. s s s s
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