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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Question
![**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 \( h(x) = (f \circ g)(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e5efc23-a198-4327-bdc7-d2931bd5f43b%2F9f27eb78-6186-4b0c-bda3-0d23d7b55527%2Fbqwwnee_processed.jpeg&w=3840&q=75)
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:
\[ f(u) = u^3 - 5u^2 + 2u - 1 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e5efc23-a198-4327-bdc7-d2931bd5f43b%2F9f27eb78-6186-4b0c-bda3-0d23d7b55527%2Fa4ibcj_processed.jpeg&w=3840&q=75)
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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