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 \((f \circ g)(4)\) for the following functions:
\[ f(x) = x^{\frac{3}{2}} \quad \text{and} \quad g(x) = 3x - 8 \]
### Solution
To find \((f \circ g)(4)\), we need to follow these steps:
1. **Calculate** \( g(4) \):
\[
g(4) = 3(4) - 8 = 12 - 8 = 4
\]
2. **Substitute** the result from step 1 into \( f(x) \):
\[
f(g(4)) = f(4)
\]
Since \( f(x) = x^{\frac{3}{2}} \):
\[
f(4) = 4^{\frac{3}{2}}
\]
To calculate \( 4^{\frac{3}{2}} \):
\[
4^{\frac{3}{2}} = (4^1) \times \sqrt{4} = 8
\]
Thus, \((f \circ g)(4) = 8\).
### Explanation
In this problem, we are dealing with the composition of two functions, where \( f(x) \) and \( g(x) \) are given. The notation \((f \circ g)(4)\) means that we first apply the function \( g \) to the number 4, and then apply the function \( f \) to the result.
Breaking it down:
1. **Function \( g \)**: This linear function transforms its input by multiplying by 3 and then subtracting 8.
2. **Function \( f \)**: This radical function raises its input to the power of \(\frac{3}{2}\).
By calculating \( g \)'s output when the input is 4, and then using that output as the input to \( f \), we find the final value.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbdfddc95-a6c9-4512-b29d-be575b082366%2F32aaf4d1-c5a2-4eb5-a6d3-579129d3d347%2Ftpaxg7k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Find \((f \circ g)(4)\) for the following functions:
\[ f(x) = x^{\frac{3}{2}} \quad \text{and} \quad g(x) = 3x - 8 \]
### Solution
To find \((f \circ g)(4)\), we need to follow these steps:
1. **Calculate** \( g(4) \):
\[
g(4) = 3(4) - 8 = 12 - 8 = 4
\]
2. **Substitute** the result from step 1 into \( f(x) \):
\[
f(g(4)) = f(4)
\]
Since \( f(x) = x^{\frac{3}{2}} \):
\[
f(4) = 4^{\frac{3}{2}}
\]
To calculate \( 4^{\frac{3}{2}} \):
\[
4^{\frac{3}{2}} = (4^1) \times \sqrt{4} = 8
\]
Thus, \((f \circ g)(4) = 8\).
### Explanation
In this problem, we are dealing with the composition of two functions, where \( f(x) \) and \( g(x) \) are given. The notation \((f \circ g)(4)\) means that we first apply the function \( g \) to the number 4, and then apply the function \( f \) to the result.
Breaking it down:
1. **Function \( g \)**: This linear function transforms its input by multiplying by 3 and then subtracting 8.
2. **Function \( f \)**: This radical function raises its input to the power of \(\frac{3}{2}\).
By calculating \( g \)'s output when the input is 4, and then using that output as the input to \( f \), we find the final value.
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