274 CHAPTER 5 Exponential and Logarithmic Functions Figure 3 provides a second illustration of the definition. Here x is the input to the function g, yielding g x( ). Then g x( ) is the input to the function f, yielding f g x ( ) ( ) . Note that the “inside” function g in f g x ( ) ( ) is “processed” first. Evaluating a Composite Function Suppose that f x x2 3 2 ( ) = − and g x x4 . ( ) = Find: (a) f g 1 ( )( ) (b) g f 1 ( )( ) (c) f f 2 ( )( ) − (d) g g 1 ( )( ) − EXAMPLE 1 (a) f g f g f 1 1 4 2 4 3 29 2 ( ) ( ) ( ) ( ) ( ) = = = ⋅ − = ↑ ↑ ( ) ( ) = = − g x x f x x 4 2 3 2 ( ) = g 1 4 (b) g f g f g 1 1 1 4 1 4 ( ) ( ) ( ) ( ) ( ) ( ) = = − = ⋅ − = − ↑ ↑ ( ) ( ) = − = f x x g x x 2 3 4 2 ( ) =− f 1 1 (c) f f f f f 2 2 5 2 5 3 47 2 ( ) ( ) ( ) ( ) ( ) − = − = = ⋅ − = ↑ ( ) ( ) − = − − = f 222 35 2 (d) g g g g g 1 1 4 4 4 16 ( ) ( ) ( ) ( ) ( ) ( ) −= − =−=⋅−=− ↑ ( ) − =− g 1 4 COMMENT Graphing utilities can evaluate composite functions.* Using a TI-84 Plus CE graphing calculator, let Y f x x2 3 1 2 ( ) = = − and Y g x x4 , 2 ( ) = = and find f g ( )(1) as shown in Figure 4(a). Using Desmos, find f g ( )(1) as shown in Figure 4(b). Note that these give the result obtained in Example 1(a). ■ Figure 3 INPUT x OUTPUT f(g(x)) f g g(x) Solution (a) TI-84 Plus CE (b) Desmos Figure 4 Now Work PROBLEM 13 2 Find the Domain of a Composite Function Finding a Composite Function and Its Domain Suppose that f x x x3 1 2 ( ) = + − and g x x2 3. ( ) = + Find: (a) f g (b) g f Then find the domain of each composite function. EXAMPLE 2 Solution The domain of f and the domain of g are the set of all real numbers. (a) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = + = + + + − = + + + + − = + + f g x f g x f x x x x x x x x 2 3 2 3 3 2 3 1 4 12 9 6 9 1 4 18 17 2 2 2 Because the domains of both f and g are the set of all real numbers, the domain of f g is the set of all real numbers. ↑ ↑ ( ) ( ) = + = + − g x x f x x x 2 3 3 1 2 *Consult your user’s manual for the appropriate keystrokes.
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