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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![Here is the transcription of the image for educational purposes:
---
**Problem Statement:**
Find \( h(x) \) and \( g(x) \) such that \( f(x) = (h \circ g)(x) \).
Given:
\[ f(x) = \sqrt{3x + 4} \]
Suppose:
\[ g(x) = 3x + 4 \]
**Task:**
Determine \( h(x) \).
**Solution:**
To find \( h(x) \), notice that since \( f(x) = (h \circ g)(x) = h(g(x)) \), we can express \( h \) by considering what \( h(x) \) must be in order to satisfy the equation.
Given \( g(x) = 3x + 4 \), substitute this into \( f(x) \):
\[ f(x) = h(g(x)) = \sqrt{3x + 4} \]
Since \( g(x) = 3x + 4 \), let \( u = g(x) = 3x + 4 \). Therefore, we want:
\[ h(u) = \sqrt{u} \]
Thus, \( h(x) = \sqrt{x} \).
**Conclusion:**
\[ h(x) = \sqrt{x} \]
This solves the problem of finding the functions \( h(x) \) and \( g(x) \) such that when composed, they produce \( f(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2f259b4-cd04-4715-ba74-9a8f7fc0fecb%2F71d1f6a9-3100-4927-8cbd-16092d1965d9%2Fje9uqha_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Here is the transcription of the image for educational purposes:
---
**Problem Statement:**
Find \( h(x) \) and \( g(x) \) such that \( f(x) = (h \circ g)(x) \).
Given:
\[ f(x) = \sqrt{3x + 4} \]
Suppose:
\[ g(x) = 3x + 4 \]
**Task:**
Determine \( h(x) \).
**Solution:**
To find \( h(x) \), notice that since \( f(x) = (h \circ g)(x) = h(g(x)) \), we can express \( h \) by considering what \( h(x) \) must be in order to satisfy the equation.
Given \( g(x) = 3x + 4 \), substitute this into \( f(x) \):
\[ f(x) = h(g(x)) = \sqrt{3x + 4} \]
Since \( g(x) = 3x + 4 \), let \( u = g(x) = 3x + 4 \). Therefore, we want:
\[ h(u) = \sqrt{u} \]
Thus, \( h(x) = \sqrt{x} \).
**Conclusion:**
\[ h(x) = \sqrt{x} \]
This solves the problem of finding the functions \( h(x) \) and \( g(x) \) such that when composed, they produce \( f(x) \).
![Find \( f(x) \) and \( g(x) \) such that \( h(x) = (f \circ g)(x) \).
\[ h(x) = (1 - 8x)^2 \]
Suppose that \( g(x) = 1 - 8x \).
\[ f(x) = \boxed{} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2f259b4-cd04-4715-ba74-9a8f7fc0fecb%2F71d1f6a9-3100-4927-8cbd-16092d1965d9%2Fjfjxy77_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Find \( f(x) \) and \( g(x) \) such that \( h(x) = (f \circ g)(x) \).
\[ h(x) = (1 - 8x)^2 \]
Suppose that \( g(x) = 1 - 8x \).
\[ f(x) = \boxed{} \]
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