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...
Related questions
Question
![**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:
\[ f(x) = x^4 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc7deb2c-0862-415e-88b2-023e0d274874%2Fe419c542-18b0-4878-b880-c9f0d0a66572%2Fqm37w4_processed.png&w=3840&q=75)
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:
\[ f(x) = x^4 \]
Expert Solution
Step 1
We know the output of function "f(x)" when input is "g(x)" and the function "g(x)" itself.
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning