Suppose f(x) is a differentiable function with f(-1) =2 and f(2) =-1. The differentiable function g(x) is defined by the formula g(x) = f(f(x)).  (a) Compute g(-1) and g(2). Explain why g(x) = 0 must have at least one solution between -1 and 2.  (b) Compute g'(-1) and g'(2) in terms of values of f and f'. Verify that g'(-1) =g'(2). Explain why g''(x) = 0 must have at least one solution between -1 and 2.

College Algebra
1st Edition
ISBN:9781938168383
Author:Jay Abramson
Publisher:Jay Abramson
Chapter3: Functions
Section3.4: Composition Of Functions
Problem 4SE: How do you find the domain for the composition of two functions, fg ?
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Suppose f(x) is a differentiable function with f(-1) =2 and f(2) =-1. The differentiable function g(x) is defined by the formula g(x) = f(f(x)). 

(a) Compute g(-1) and g(2). Explain why g(x) = 0 must have at least one solution between -1 and 2. 

(b) Compute g'(-1) and g'(2) in terms of values of f and f'. Verify that g'(-1) =g'(2). Explain why g''(x) = 0 must have at least one solution between -1 and 2.

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