SECTION 5.1 Composite Functions 277 3. Find the domain of the function f x x x 1 25 . 2 2 ( ) = − − (pp. 69–70) 1. Find f 3( ) if f x x x 4 5 . 2 ( ) = − + (pp. 65–67) 2. Find f x3( ) if f x x 4 2 .2 ( ) = − (pp. 65–67) ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. In Section 5.2, we shall see that there is an important relationship between functions f and g for which f g x g f x x. ( ) ( ) ( ) ( ) = = Now Work PROBLEM 39 Calculus Application Some techniques in calculus require the ability to determine the components of a composite function. For example, the function H x x 1 ( ) = + is the composition of the functions f and g , where f x x ( ) = and g x x 1, ( ) = + because H x f g x f g x f x x 1 1. ( ) ( ) ( ) ( ) ( ) ( ) = = = + = + Finding the Components of a Composite Function Find functions f and g so that f g H = when H x x 1 . 2 50 ( ) ( ) = + EXAMPLE 6 The function H raises the expression x 1 2 + to the power 50. A natural way to decompose H is to raise the function g x x 1 2 ( ) = + to the power 50. Let f x x50 ( ) = and g x x 1. 2 ( ) = + Then f g x f g x f x x H x 1 1 2 2 50 ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = + = + = See Figure 5. Solution Figure 5 x H g f f(g(x)) = f(x2 + 1) H(x) = (x2 + 1)50 g(x) = x2 + 1 = (x2 + 1)50 Other functions f and g may be found for which f g H = in Example 6. For instance, if f x x2 ( ) = and g x x 1 , 2 25 ( ) ( ) = + then f g x f g x f x x x 1 1 1 2 25 2 25 2 2 50 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = + = ⎡ ⎣ + ⎤⎦ = + Although the functions f and g found as a solution to Example 6 are not unique, there is usually a “natural” selection for f and g that comes to mind first. Finding the Components of a Composite Function Find functions f and g so that f g H = when H x x 1 1 . ( ) = + Solution EXAMPLE 7 Here H is the reciprocal of g x x 1. ( ) = + Let f x x 1 ( ) = and g x x 1. ( ) = + Then f g x f g x f x x H x 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) = = + = + = Now Work PROBLEM 47 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure 5.1 Assess Your Understanding
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