Find f(x) and g(x) such that h(x) = (f o g)(x) and g(x) =/x +1. h(x) = (\/x + 1)*

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 \( f(x) \) and \( g(x) \) such that \( h(x) = (f \circ g)(x) \) and \( g(x) = \sqrt{x} + 1 \).

\[ h(x) = \left( \sqrt{x} + 1 \right)^4 \]

**Solution:**

To solve this problem, we need to identify functions \( f(x) \) and \( g(x) \) that satisfy the conditions given.

Given:

1. \( g(x) = \sqrt{x} + 1 \)

2. \( h(x) = (f \circ g)(x) \)

This implies:

\[ h(x) = f(g(x)) = \left( \sqrt{x} + 1 \right)^4 \]

Since \( g(x) = \sqrt{x} + 1 \), we substitute \( g(x) \) into \( h(x) \):

\[ f(g(x)) = f(\sqrt{x} + 1) = \left( \sqrt{x} + 1 \right)^4 \]

To determine \( f(x) \), observe that if we let:

\[ u = \sqrt{x} + 1 \]

Then \( h(x) = u^4 \).

Therefore, \( f(u) = u^4 \).

So:

\[ f(x) = x^4 \]

Thus, the functions are:

\[ g(x) = \sqrt{x} + 1 \]
\[ f(x) = x^4 \]

These functions satisfy the condition \( h(x) = (f \circ g)(x) \).
Transcribed Image Text:**Problem Statement:** Find \( f(x) \) and \( g(x) \) such that \( h(x) = (f \circ g)(x) \) and \( g(x) = \sqrt{x} + 1 \). \[ h(x) = \left( \sqrt{x} + 1 \right)^4 \] **Solution:** To solve this problem, we need to identify functions \( f(x) \) and \( g(x) \) that satisfy the conditions given. Given: 1. \( g(x) = \sqrt{x} + 1 \) 2. \( h(x) = (f \circ g)(x) \) This implies: \[ h(x) = f(g(x)) = \left( \sqrt{x} + 1 \right)^4 \] Since \( g(x) = \sqrt{x} + 1 \), we substitute \( g(x) \) into \( h(x) \): \[ f(g(x)) = f(\sqrt{x} + 1) = \left( \sqrt{x} + 1 \right)^4 \] To determine \( f(x) \), observe that if we let: \[ u = \sqrt{x} + 1 \] Then \( h(x) = u^4 \). Therefore, \( f(u) = u^4 \). So: \[ f(x) = x^4 \] Thus, the functions are: \[ g(x) = \sqrt{x} + 1 \] \[ f(x) = x^4 \] These functions satisfy the condition \( h(x) = (f \circ g)(x) \).
**Problem Statement:**

Find \( f(x) \) and \( g(x) \) such that \( h(x) = (f \circ g)(x) \) and \( g(x) = 6 - 8x \).

**Given:**

\[ h(x) = (6 - 8x)^3 - 5(6 - 8x)^2 + 2(6 - 8x) - 1 \]

**Solution:**

First, note that the function \( g(x) \) is already provided as:

\[ g(x) = 6 - 8x \]

Next, identify \( f(u) \) such that when \( u = g(x) \), \( f(u) \) matches the structure of \( h(x) \).

From the form of \( h(x) \), let \( u = 6 - 8x \), then:

\[ f(u) = u^3 - 5u^2 + 2u - 1 \]

Thus, \( f(x) \) is defined as:

\[ f(u) = u^3 - 5u^2 + 2u - 1 \]
Transcribed Image Text:**Problem Statement:** Find \( f(x) \) and \( g(x) \) such that \( h(x) = (f \circ g)(x) \) and \( g(x) = 6 - 8x \). **Given:** \[ h(x) = (6 - 8x)^3 - 5(6 - 8x)^2 + 2(6 - 8x) - 1 \] **Solution:** First, note that the function \( g(x) \) is already provided as: \[ g(x) = 6 - 8x \] Next, identify \( f(u) \) such that when \( u = g(x) \), \( f(u) \) matches the structure of \( h(x) \). From the form of \( h(x) \), let \( u = 6 - 8x \), then: \[ f(u) = u^3 - 5u^2 + 2u - 1 \] Thus, \( f(x) \) is defined as: \[ f(u) = u^3 - 5u^2 + 2u - 1 \]
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