To solve for $f(g(x))$, let's first clarify the functions given:

1. $f(x) = \frac{1}{x - 5}$
2. $g(x) = \frac{1}{x} - 5$

Step 1: Substitute $g(x)$ into $f(x)$

Since we want $f(g(x))$, we substitute $g(x)$ into $f$: $f(g(x)) = f\left(\frac{1}{x} - 5\right)$

Step 2: Substitute into $f(x) = \frac{1}{x - 5}$

Replace $x$ in $f(x)$ with $g(x) = \frac{1}{x} - 5$: $f(g(x)) = \frac{1}{\left(\frac{1}{x} - 5\right) - 5}$

Step 3: Simplify the Expression

Now simplify the expression inside the denominator: $f(g(x)) = \frac{1}{\frac{1}{x} - 5 - 5}$ $= \frac{1}{\frac{1}{x} - 10}$

To further simplify: $f(g(x)) = \frac{1}{\frac{1 - 10x}{x}} = \frac{x}{1 - 10x}$

So, $f(g(x)) = \frac{x}{1 - 10x}$

Answer:

$f(g(x)) = \frac{x}{1 - 10x}$

Would you like more details on the simplification steps?

Here are some related questions to explore further:

1. How would you calculate $g(f(x))$ instead of $f(g(x))$?
2. What are the domains of $f(x)$ and $g(x)$, and how do they impact the domain of $f(g(x))$?
3. How would you find $f(f(x))$ or $g(g(x))$?
4. Can you determine the inverse functions for $f(x)$ and $g(x)$?
5. What happens to $f(g(x))$ as $x$ approaches certain values that make the function undefined?

Tip: When composing functions, always consider the domain restrictions that may affect the final result.