305 2.8 Function Operations and Composition (b) 1g∘ ƒ21x2 = g1ƒ1x22 By definition = g a 6 x - 3b ƒ1x2 = 6 x - 3 = 1 6 x - 3 Note that this is meaningless if x = 3; g1x2 = 1 x = x - 3 6 1 a b = 1 , a b = 1 # b a = b a The domain of ƒ is the set of all real numbers except 3. The domain of g is the set of all real numbers except 0. The expression for ƒ1x2, which is 6 x - 3 , is never 0 because the numerator is the nonzero number 6. The domain of g∘ ƒ is the set of all real numbers except 3, written 1-∞, 32 ´13, ∞2. S Now Try Exercise 87. LOOKING AHEAD TO CALCULUS Finding the derivative of a function in calculus is called differentiation. To differentiate a composite function such as h1x2 = 13x + 224, we interpret h1x2 as 1ƒ ∘ g21x2, where g1x2 = 3x + 2 and ƒ1x2 = x4. The chain rule allows us to differentiate composite functions. Notice the use of the composition symbol and function notation in the following, which comes from the chain rule. If h1x2 = 1ƒ ∘ g21x2, then h′1x2 = ƒ′1g1x22 # g′1x2. NOTE It often helps to consider the unsimplified form of a composition expression when determining the domain in a situation like Example 7(b). As Example 8 shows, it is not always true that ƒ° g =g° ƒ. One important circumstance in which equality holds occurs when ƒ and g are inverses of each other, a concept discussed later in the text. EXAMPLE 8 Showing That 1g° f 2 1x2 Is Not Equivalent to 1f ° g2 1x2 Let ƒ1x2 = 4x + 1 and g1x2 = 2x2 + 5x. Show that 1g∘ ƒ21x2 ≠1ƒ∘ g21x2. (This is sufficient to prove that this inequality is true in general.) SOLUTION First, find 1g∘ ƒ21x2. Then find 1ƒ∘ g21x2. 1g∘ ƒ21x2 = g1ƒ1x22 By definition = g14x + 12 ƒ1x2 = 4x + 1 = 214x + 122 + 514x + 12 g1x2 = 2x2 + 5x = 2116x2 + 8x + 12 + 20x + 5 Square 4x + 1 and apply the distributive property. = 32x2 + 16x + 2 + 20x + 5 Distributive property = 32x2 + 36x + 7 Combine like terms. 1ƒ∘ g21x = ƒ1g1x22 By definition = ƒ12x2 + 5x2 g1x2 = 2x2 + 5x = 412x2 + 5x2 + 1 ƒ1x2 = 4x + 1 = 8x2 + 20x + 1 Distributive property Thus, 1g∘ ƒ21x2 ≠1ƒ∘ g21x2. S Now Try Exercise 91.
RkJQdWJsaXNoZXIy NjM5ODQ=