Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
ChapterP: Prerequisites
SectionP.5: Functions
Problem 9ECP
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Question
![### Composite Functions Problem
Given two functions:
\[ f(x) = \frac{1}{x - 3} \]
\[ g(x) = \frac{5}{x} + 3 \]
### Task:
Find the following composite functions and simplify your answers.
1. \( f(g(x)) = \)
2. \( g(f(x)) = \)
### Solution Steps:
#### Finding \( f(g(x)) \):
Here, we need to substitute \( g(x) \) into \( f(x) \). First, identify the value of \( g(x) \):
\[ g(x) = \frac{5}{x} + 3 \]
Then, substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f\left(\frac{5}{x} + 3\right) \]
\[ f\left(\frac{5}{x} + 3\right) = \frac{1}{\left(\frac{5}{x} + 3\right) - 3} \]
Simplify the expression inside the denominator:
\[ \left(\frac{5}{x} + 3\right) - 3 = \frac{5}{x} \]
Thus:
\[ f(g(x)) = f\left(\frac{5}{x} + 3\right) = \frac{1}{\frac{5}{x}} \]
Simplify further:
\[ f(g(x)) = \frac{1 \cdot x}{5} = \frac{x}{5} \]
#### Finding \( g(f(x)) \):
Now, we need to substitute \( f(x) \) into \( g(x) \). First, identify the value of \( f(x) \):
\[ f(x) = \frac{1}{x - 3} \]
Then, substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g\left(\frac{1}{x - 3}\right) \]
\[ g\left(\frac{1}{x - 3}\right) = \frac{5}{\frac{1}{x - 3}} + 3 \]
Simplify the fraction in the first term:
\[ \frac{5}{\frac{1}{x - 3}} = 5 \cdot (x - 3) = 5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88d18a19-21f4-41fb-afe0-306734676977%2F16b042af-7742-4aad-b347-8eab1e9bd77a%2Ft7lxiqk.png&w=3840&q=75)
Transcribed Image Text:### Composite Functions Problem
Given two functions:
\[ f(x) = \frac{1}{x - 3} \]
\[ g(x) = \frac{5}{x} + 3 \]
### Task:
Find the following composite functions and simplify your answers.
1. \( f(g(x)) = \)
2. \( g(f(x)) = \)
### Solution Steps:
#### Finding \( f(g(x)) \):
Here, we need to substitute \( g(x) \) into \( f(x) \). First, identify the value of \( g(x) \):
\[ g(x) = \frac{5}{x} + 3 \]
Then, substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f\left(\frac{5}{x} + 3\right) \]
\[ f\left(\frac{5}{x} + 3\right) = \frac{1}{\left(\frac{5}{x} + 3\right) - 3} \]
Simplify the expression inside the denominator:
\[ \left(\frac{5}{x} + 3\right) - 3 = \frac{5}{x} \]
Thus:
\[ f(g(x)) = f\left(\frac{5}{x} + 3\right) = \frac{1}{\frac{5}{x}} \]
Simplify further:
\[ f(g(x)) = \frac{1 \cdot x}{5} = \frac{x}{5} \]
#### Finding \( g(f(x)) \):
Now, we need to substitute \( f(x) \) into \( g(x) \). First, identify the value of \( f(x) \):
\[ f(x) = \frac{1}{x - 3} \]
Then, substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g\left(\frac{1}{x - 3}\right) \]
\[ g\left(\frac{1}{x - 3}\right) = \frac{5}{\frac{1}{x - 3}} + 3 \]
Simplify the fraction in the first term:
\[ \frac{5}{\frac{1}{x - 3}} = 5 \cdot (x - 3) = 5
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