Sure, let's go through the provided mathematical functions step-by-step to answer the question.
We are given the functions:
[tex]\[
f(x) = 5x
\][/tex]
[tex]\[
g(x) = \frac{x}{5}
\][/tex]
### Step 1: Finding [tex]\( f(g(x)) \)[/tex]
First, let's find the composite function [tex]\( f(g(x)) \)[/tex]. This means we will substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex].
Given:
[tex]\[
g(x) = \frac{x}{5}
\][/tex]
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(g(x)) = f\left( \frac{x}{5} \right)
\][/tex]
Now apply the function [tex]\( f \)[/tex]:
[tex]\[
f\left( \frac{x}{5} \right) = 5 \left( \frac{x}{5} \right)
\][/tex]
Simplify the expression:
[tex]\[
f\left( \frac{x}{5} \right) = x
\][/tex]
So:
[tex]\[
f(g(x)) = x
\][/tex]
### Step 2: Finding [tex]\( g(f(x)) \)[/tex]
Next, we will find the composite function [tex]\( g(f(x)) \)[/tex]. This means we will substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex].
Given:
[tex]\[
f(x) = 5x
\][/tex]
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(f(x)) = g(5x)
\][/tex]
Now apply the function [tex]\( g \)[/tex]:
[tex]\[
g(5x) = \frac{5x}{5}
\][/tex]
Simplify the expression:
[tex]\[
g(5x) = x
\][/tex]
So:
[tex]\[
g(f(x)) = x
\][/tex]
### Step 3: Determining if [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are Inverses
To determine whether [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses of each other, both of the following must be true:
[tex]\[
f(g(x)) = x
\][/tex]
[tex]\[
g(f(x)) = x
\][/tex]
From the calculations, we have:
[tex]\[
f(g(x)) = x
\][/tex]
[tex]\[
g(f(x)) = x
\][/tex]
Both conditions are satisfied.
Therefore, [tex]\( f(x) = 5x \)[/tex] and [tex]\( g(x) = \frac{x}{5} \)[/tex] are indeed inverses of each other.
