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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![# Evaluating a Composite Function
To evaluate the given composite function \( h(4) \), where \( h = g \circ f \), follow these steps:
### Given Functions:
\[ f(x) = \sqrt[3]{x^2 - 9} \]
\[ g(x) = 7x^3 + 9 \]
### Objective:
Evaluate \( h(4) \).
### Steps:
1. **Find \( f(4) \):**
Substitute \( x = 4 \) into the function \( f(x) = \sqrt[3]{x^2 - 9} \):
\[
f(4) = \sqrt[3]{4^2 - 9} = \sqrt[3]{16 - 9} = \sqrt[3]{7}
\]
2. **Find \( g(f(4)) \):**
Next, substitute \( f(4) \) into \( g(x) = 7x^3 + 9 \):
\[
g(f(4)) = g\left(\sqrt[3]{7}\right) = 7\left(\sqrt[3]{7}\right)^3 + 9
\]
Since \( \left(\sqrt[3]{7}\right)^3 = 7 \):
\[
g(f(4)) = 7 \times 7 + 9 = 49 + 9 = 58
\]
Thus, the value of \( h(4) \) is:
\[
h(4) = 58
\]
### Conclusion:
By evaluating the composite function step-by-step, we found that \( h(4) = 58 \). This method can be applied to other composite functions to determine their values at specific points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8eeb6921-a110-4984-a04e-b3e01fc97c1d%2F039910b8-5557-48e1-abd5-b9e4ac53e3a7%2F542jn0t9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Evaluating a Composite Function
To evaluate the given composite function \( h(4) \), where \( h = g \circ f \), follow these steps:
### Given Functions:
\[ f(x) = \sqrt[3]{x^2 - 9} \]
\[ g(x) = 7x^3 + 9 \]
### Objective:
Evaluate \( h(4) \).
### Steps:
1. **Find \( f(4) \):**
Substitute \( x = 4 \) into the function \( f(x) = \sqrt[3]{x^2 - 9} \):
\[
f(4) = \sqrt[3]{4^2 - 9} = \sqrt[3]{16 - 9} = \sqrt[3]{7}
\]
2. **Find \( g(f(4)) \):**
Next, substitute \( f(4) \) into \( g(x) = 7x^3 + 9 \):
\[
g(f(4)) = g\left(\sqrt[3]{7}\right) = 7\left(\sqrt[3]{7}\right)^3 + 9
\]
Since \( \left(\sqrt[3]{7}\right)^3 = 7 \):
\[
g(f(4)) = 7 \times 7 + 9 = 49 + 9 = 58
\]
Thus, the value of \( h(4) \) is:
\[
h(4) = 58
\]
### Conclusion:
By evaluating the composite function step-by-step, we found that \( h(4) = 58 \). This method can be applied to other composite functions to determine their values at specific points.
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