To find the function $f(g(x))$, we need to substitute $g(x)$ into the function $f(x)$.

Given: $f(x) = \frac{x}{x - 7}$ $g(x) = \frac{3}{x}$

We substitute $g(x)$ into $f(x)$:

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

Now, replace $x$ in $f(x)$ with $\frac{3}{x}$:

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

Next, simplify the expression:

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

Simplify further:

$f(g(x)) = \frac{3/x}{(3 - 7x)/x} = \frac{3}{3 - 7x}$

So, the function $f(g(x))$ is:

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

Would you like more details or have any questions about this process?

Here are some related questions:

1. How do you find the domain of $f(g(x))$?
2. What is $g(f(x))$?
3. How do you determine the inverse of $f(x)$?
4. What happens if $x = 7$ in $f(x)$?
5. How would you graph $f(g(x))$?

Tip: When substituting functions, ensure all instances of the variable are replaced correctly to avoid mistakes.