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We are asked to find \( (f \circ g)(x) \) and \( (g \circ f)(x) \), given:

- \( f(x) = 5x - 4 \)
- \( g(x) = \frac{1}{5}(x + 4) \)

Then we need to determine if \( (f \circ g)(x) = (g \circ f)(x) \).

### Step 1: Finding \( (f \circ g)(x) \)
The composition \( (f \circ g)(x) \) means applying \( g(x) \) first and then applying \( f \) to the result of \( g(x) \).

\[
(f \circ g)(x) = f(g(x)) = f\left(\frac{1}{5}(x + 4)\right)
\]

Now, substitute \( g(x) = \frac{1}{5}(x + 4) \) into \( f(x) = 5x - 4 \):

\[
f\left(\frac{1}{5}(x + 4)\right) = 5\left(\frac{1}{5}(x + 4)\right) - 4 = (x + 4) - 4 = x
\]

So,
\[
(f \circ g)(x) = x
\]

### Step 2: Finding \( (g \circ f)(x) \)
The composition \( (g \circ f)(x) \) means applying \( f(x) \) first and then applying \( g \) to the result of \( f(x) \).

\[
(g \circ f)(x) = g(f(x)) = g(5x - 4)
\]

Now, substitute \( f(x) = 5x - 4 \) into \( g(x) = \frac{1}{5}(x + 4) \):

\[
g(5x - 4) = \frac{1}{5}((5x - 4) + 4) = \frac{1}{5}(5x) = x
\]

So,
\[
(g \circ f)(x) = x
\]

### Step 3: Conclusion
Since \( (f \circ g)(x) = x \) and \( (g \circ f)(x) = x \), we can conclude that:

\[
(f \circ g)(x) = (g \circ f)(x)
\]

Would you like further details or explanations? Here are five related questions:

1. What are some properties of function composition?
2. Can a composition of functions ever be undefined?
3. How do you verify if two compositions of functions are equal in general?
4. How would you approach a similar problem with different function types?
5. What happens if you change the order of composition for non-linear functions?

**Tip:** Always verify the domain of each function before composing them to ensure the composition is well-defined.

For f(x) = 5x - 4 and g(x) = (1/5)(x + 4), find (f o g)(x) and (g o f)(x). Then determine whether (f o g)(x) = (g o f)(x).

Solve the composition of two linear functions: f(x) = 5x - 4 and g(x) = (1/5)(x + 4). Find (f o g)(x) and (g o f)(x), and determine if they are equal. This problem covers function composition and commutativity for linear functions, suitable for high school and college algebra students.

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