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Find f(x) and g(x) such that h(x) = (f o g)(x) and g(x) = x +8. h(x) = (/x + 8) * *+*)* f(x) =O

Find f(x) and g(x) such that h(x) = (f o g)(x) and g(x) = x +8. h(x) = (/x + 8) * *+*)* f(x) =O

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem Statement:** Find functions \( f(x) \) and \( g(x) \) such that \( h(x) = (f \circ g)(x) \) and \( g(x) = \sqrt{x} + 8 \). Given: \[ h(x) = \left( \sqrt{x} + 8 \right)^4 \] **Solution Part:** Determine the function \( f(x) \): Evaluate: Since \( h(x) = (f \circ g)(x) \), this implies \( h(x) = f(g(x)) \). Substitute the given \( g(x) \): \[ g(x) = \sqrt{x} + 8 \] Substitute \( g(x) \) in \( f(g(x)) \): \[ h(x) = f\left(\sqrt{x} + 8\right) = \left( \sqrt{x} + 8 \right)^4 \] Thus, \( f(u) = u^4 \), where \( u = g(x) = \sqrt{x} + 8 \). Therefore, the function \( f(x) \) is: \[ f(x) = x^4 \]
Transcribed Image Text:**Problem Statement:** Find functions \( f(x) \) and \( g(x) \) such that \( h(x) = (f \circ g)(x) \) and \( g(x) = \sqrt{x} + 8 \). Given: \[ h(x) = \left( \sqrt{x} + 8 \right)^4 \] **Solution Part:** Determine the function \( f(x) \): Evaluate: Since \( h(x) = (f \circ g)(x) \), this implies \( h(x) = f(g(x)) \). Substitute the given \( g(x) \): \[ g(x) = \sqrt{x} + 8 \] Substitute \( g(x) \) in \( f(g(x)) \): \[ h(x) = f\left(\sqrt{x} + 8\right) = \left( \sqrt{x} + 8 \right)^4 \] Thus, \( f(u) = u^4 \), where \( u = g(x) = \sqrt{x} + 8 \). Therefore, the function \( f(x) \) is: \[ f(x) = x^4 \]
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We know the output of function "f(x)" when input is "g(x)" and the function "g(x)" itself.

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