299 2.8 Function Operations and Composition (c) 1ƒg21x2 = ƒ1x2 # g1x2 = 18x - 9222x - 1 (d) a ƒ gb1x2 = ƒ1x2 g1x2 = 8x - 9 2 2x - 1 (e) To find the domains of the functions in parts (a)–(d), we first find the domains of ƒ and g. The domain of ƒ is the set of all real numbers 1-∞, ∞2. Because g is defined by a square root radical, the radicand must be nonnegative—that is, greater than or equal to 0. g1x2 = 22x - 1 Rule for g1x2 2x - 1 Ú 0 2x - 1 must be nonnegative. x Ú 1 2 Add 1 and divide by 2. Thus, the domain of g is C 1 2 , ∞B. The domains of ƒ + g, ƒ - g, and ƒg are the intersection of the domains of ƒ and g. 1-∞, ∞2 ¨c 1 2 , ∞b = c 1 2 , ∞b The intersection of two sets is the set of all elements common to both sets. The domain of ƒ g includes those real numbers in the intersection above for which g1x2 = 22x - 1≠0—that is, the domain of ƒ g is A 1 2 , ∞B. S Now Try Exercise 19. EXAMPLE 3 Evaluating Combinations of Functions If possible, use the given representations of functions ƒ and g in parts (a)–(c) to evaluate the following. 1ƒ + g2142, 1ƒ - g21-22, 1ƒg2112, and a f gb102 (a) –4 –2 1 2 4 –3 –1 1 5 9 x y y = f(x) y = g(x) (b) x ƒ1x2 g1x2 -2 -3 undefined 0 1 0 1 3 1 4 9 2 (c) ƒ1x2 = 2x + 1, g1x2 = 2x
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