320 CHAPTER 2 Graphs and Functions (Modeling) Solve each problem. 127. Relationship of Measurement Units There are 36 in. in 1 yd, and there are 1760 yd in 1 mi. Express the number of inches in x miles by forming two functions and then considering their composition. 128. Perimeter of a Rectangle The length of a rectangle is twice its width. Let x represent the width. Write a formula for the perimeter P in terms of x alone. Then use P1x2 notation to describe it as a function. What type of function is this? 129. Volume of a Sphere The formula for the volume of a sphere is V1r2 = 4 3pr 3, where r represents the radius. Construct a model function V representing the amount of volume gained when the radius r (in inches) is increased by 3 in. 130. Dimensions of a Cylinder A cylindrical can has height the same as the diameter of its top. (a) Express the volume V of such a can as a function of the diameter d of its top. (b) Express the surface area S of such a can as a function of the diameter d of its top. (Hint: The curved side is made from a rectangle whose length is the circumference of the top of the can.) For each function, find and simplify ƒ1x + h2 - ƒ1x2 h , h≠0. 111. ƒ1x2 = 2x + 9 112. ƒ1x2 = x2 - 5x + 3 Use the tables for ƒ and g to evaluate each expression. x ƒ1x2 -2 1 0 4 2 3 4 2 x g1x2 1 2 2 4 3 -2 4 0 123. 1g∘ ƒ21-22 124. 1ƒ∘ g2132 r d Concept Check The graphs of two functions ƒ and g are shown in the figures. 125. Find 1ƒ∘ g2122. 126. Find 1g∘ ƒ2132. 4 1 0 2 (3, 4) (2, 1) y = f(x) x y 8 0 (1, –1) y = g(x) x y (2, 2) 4 (4, 8) x ƒ1x2 g1x2 -1 3 -2 0 5 0 1 7 1 3 9 9 120. 1ƒ - g2132 121. 1ƒg21-12 122. a ƒ gb102 Use the table to evaluate each expression, if possible. 119. 1ƒ + g2112 Let ƒ1x2 = 2x - 2 and g1x2 = x2. Find each of the following, if possible. 113. 1g∘ ƒ21x2 114. 1ƒ∘ g21x2 115. 1g∘ ƒ2132 116. 1ƒ∘ g21-62 117. 1g∘ ƒ21-12 118. the domain of ƒ∘ g
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