Evaluate (fog)(x) and write the domain in interval notation. Write the answer in the intervals as an integer or simplified fraction. f(x)=x²-1 Part: 0/2 Part 1 of 2 (fog)(x) = 11 ²-25 X 0² S Es O G E 9
Evaluate (fog)(x) and write the domain in interval notation. Write the answer in the intervals as an integer or simplified fraction. f(x)=x²-1 Part: 0/2 Part 1 of 2 (fog)(x) = 11 ²-25 X 0² S Es O G E 9
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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![**Composition of Functions: Evaluate \( (f \circ g)(x) \) and Determine the Domain**
Given the functions:
\[ f(x) = \frac{x}{x - 1} \]
\[ g(x) = \frac{11}{x^2 - 25} \]
**Task:**
Evaluate \( (f \circ g)(x) \) and write the domain in interval notation. Write the answer in the intervals as an integer or simplified fraction.
### Step-by-Step Solution
1. **Compose the Functions:**
\[ (f \circ g)(x) = f(g(x)) \]
Which means substituting \( g(x) \) into \( f(x) \).
\[ g(x) = \frac{11}{x^2 - 25} \]
Substitute \( g(x) \) into \( f(x) \):
\[ (f \circ g)(x) = f\left( \frac{11}{x^2 - 25} \right) \]
2. **Evaluate the Expression:**
Substitute \( \frac{11}{x^2 - 25} \) into \( f(x) \)'s form \( \frac{x}{x - 1} \):
\[ f(g(x)) = \frac{\frac{11}{x^2 - 25}}{\frac{11}{x^2 - 25} - 1} \]
Simplify the denominator:
\[ \frac{11}{x^2 - 25} - 1 = \frac{11 - (x^2 - 25)}{x^2 - 25} = \frac{11 - x^2 + 25}{x^2 - 25} = \frac{36 - x^2}{x^2 - 25} \]
Therefore:
\[ (f \circ g)(x) = \frac{\frac{11}{x^2 - 25}}{\frac{36 - x^2}{x^2 - 25}} = \frac{11}{36 - x^2} \]
3. **Domain:**
To find the domain of \( (f \circ g)(x) \), we need to consider the restrictions of both \( f(x) \) and \( g(x) \).
- First, \( g(x) \) must be defined:
\[ x^2 - 25 \neq](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d66a6c9-1e31-4be5-aa32-733147e3a333%2F59102a5f-a509-4761-b103-9e5a26ef3e83%2Ftwtc58j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Composition of Functions: Evaluate \( (f \circ g)(x) \) and Determine the Domain**
Given the functions:
\[ f(x) = \frac{x}{x - 1} \]
\[ g(x) = \frac{11}{x^2 - 25} \]
**Task:**
Evaluate \( (f \circ g)(x) \) and write the domain in interval notation. Write the answer in the intervals as an integer or simplified fraction.
### Step-by-Step Solution
1. **Compose the Functions:**
\[ (f \circ g)(x) = f(g(x)) \]
Which means substituting \( g(x) \) into \( f(x) \).
\[ g(x) = \frac{11}{x^2 - 25} \]
Substitute \( g(x) \) into \( f(x) \):
\[ (f \circ g)(x) = f\left( \frac{11}{x^2 - 25} \right) \]
2. **Evaluate the Expression:**
Substitute \( \frac{11}{x^2 - 25} \) into \( f(x) \)'s form \( \frac{x}{x - 1} \):
\[ f(g(x)) = \frac{\frac{11}{x^2 - 25}}{\frac{11}{x^2 - 25} - 1} \]
Simplify the denominator:
\[ \frac{11}{x^2 - 25} - 1 = \frac{11 - (x^2 - 25)}{x^2 - 25} = \frac{11 - x^2 + 25}{x^2 - 25} = \frac{36 - x^2}{x^2 - 25} \]
Therefore:
\[ (f \circ g)(x) = \frac{\frac{11}{x^2 - 25}}{\frac{36 - x^2}{x^2 - 25}} = \frac{11}{36 - x^2} \]
3. **Domain:**
To find the domain of \( (f \circ g)(x) \), we need to consider the restrictions of both \( f(x) \) and \( g(x) \).
- First, \( g(x) \) must be defined:
\[ x^2 - 25 \neq
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