Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter9: Quadratic Functions And Equations
Section9.9: Combining Functions
Problem 52HP
Related questions
Question
![Title: Composite Functions
---
### Problem Statement
Given the functions:
\[ f(x) = \frac{x}{x-5} \quad \text{and} \quad g(x) = \frac{7}{x} \]
Find the following composite functions. Simplify your answers.
#### 1. \( f(g(x)) = \)
\[ \boxed{\phantom{insert answer here}} \]
#### 2. \( g(f(x)) = \)
\[ \boxed{\phantom{insert answer here}} \]
---
### Explanation
To find the composite functions \( f(g(x)) \) and \( g(f(x)) \), we will substitute the expressions for \( g(x) \) and \( f(x) \) into each other and simplify the resulting expressions.
1. **Finding \( f(g(x)) \):**
\[
f(g(x)) = f\left(\frac{7}{x}\right)
\]
Substitute \(\frac{7}{x}\) into \( f(x) \):
\[
f\left(\frac{7}{x}\right) = \frac{\frac{7}{x}}{\frac{7}{x} - 5}
\]
Simplify the expression.
2. **Finding \( g(f(x)) \):**
\[
g(f(x)) = g\left(\frac{x}{x-5}\right)
\]
Substitute \(\frac{x}{x-5}\) into \( g(x) \):
\[
g\left(\frac{x}{x-5}\right) = \frac{7}{\frac{x}{x-5}}
\]
Simplify the expression.
---
Once you have the expressions simplified, insert them into the provided boxes to complete the problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa78a88d7-60f0-4b9e-a272-6ff852667c68%2F6d09daca-62f7-45e5-ab7d-4f341769805f%2F0d2skxk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Title: Composite Functions
---
### Problem Statement
Given the functions:
\[ f(x) = \frac{x}{x-5} \quad \text{and} \quad g(x) = \frac{7}{x} \]
Find the following composite functions. Simplify your answers.
#### 1. \( f(g(x)) = \)
\[ \boxed{\phantom{insert answer here}} \]
#### 2. \( g(f(x)) = \)
\[ \boxed{\phantom{insert answer here}} \]
---
### Explanation
To find the composite functions \( f(g(x)) \) and \( g(f(x)) \), we will substitute the expressions for \( g(x) \) and \( f(x) \) into each other and simplify the resulting expressions.
1. **Finding \( f(g(x)) \):**
\[
f(g(x)) = f\left(\frac{7}{x}\right)
\]
Substitute \(\frac{7}{x}\) into \( f(x) \):
\[
f\left(\frac{7}{x}\right) = \frac{\frac{7}{x}}{\frac{7}{x} - 5}
\]
Simplify the expression.
2. **Finding \( g(f(x)) \):**
\[
g(f(x)) = g\left(\frac{x}{x-5}\right)
\]
Substitute \(\frac{x}{x-5}\) into \( g(x) \):
\[
g\left(\frac{x}{x-5}\right) = \frac{7}{\frac{x}{x-5}}
\]
Simplify the expression.
---
Once you have the expressions simplified, insert them into the provided boxes to complete the problem.
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