To solve for $f(g(x))$ and $g(f(x))$, we need to substitute one function into the other and simplify.

Given: $f(x) = 3x, \quad g(x) = \frac{x + 7}{x^3}$

1. Finding $f(g(x))$

To find $f(g(x))$, we substitute $g(x)$ into $f(x)$:

$f(g(x)) = f\left( \frac{x + 7}{x^3} \right)$

Since $f(x) = 3x$, we replace $x$ in $f(x)$ with $\frac{x + 7}{x^3}$:

$f(g(x)) = 3 \left( \frac{x + 7}{x^3} \right)$

Simplify this expression:

$f(g(x)) = \frac{3(x + 7)}{x^3}$

2. Finding $g(f(x))$

To find $g(f(x))$, we substitute $f(x)$ into $g(x)$:

$g(f(x)) = g(3x)$

Since $g(x) = \frac{x + 7}{x^3}$, we replace $x$ in $g(x)$ with $3x$:

$g(f(x)) = \frac{3x + 7}{(3x)^3}$

Now, simplify the denominator:

$g(f(x)) = \frac{3x + 7}{27x^3}$

Final Simplified Results

$f(g(x)) = \frac{3(x + 7)}{x^3}, \quad g(f(x)) = \frac{3x + 7}{27x^3}$

Would you like more details on these steps, or do you have any questions?

Here are 5 related questions that might interest you:

1. How do you compute the composition of functions in general?
2. What are some real-world applications of function compositions?
3. How do you determine the domain of a composed function?
4. What is the inverse of a composite function?
5. How do you interpret the graph of a composite function?

Tip: Remember, when composing functions, always substitute the inner function completely into the outer function and simplify!