We are asked to find $f(g(x))$, where $f(x) = \frac{1}{x+5}$ and $g(x) = \frac{x}{x-9}$.

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

$f(g(x)) = f\left(\frac{x}{x-9}\right)$ Since $f(x) = \frac{1}{x+5}$, we replace $x$ in $f(x)$ with $\frac{x}{x-9}$:

$f(g(x)) = \frac{1}{\left(\frac{x}{x-9}\right) + 5}$

Now, let's simplify the denominator:

$\left(\frac{x}{x-9}\right) + 5 = \frac{x}{x-9} + \frac{5(x-9)}{x-9} = \frac{x + 5(x-9)}{x-9}$ Simplifying further: $x + 5(x-9) = x + 5x - 45 = 6x - 45$

So the expression becomes:

$f(g(x)) = \frac{1}{\frac{6x - 45}{x-9}} = \frac{x-9}{6x-45}$

Thus, the simplified expression for $f(g(x))$ is:

$f(g(x)) = \frac{x-9}{6x-45}$

Would you like a deeper explanation or any details on the steps? Here are some related questions you might consider:

1. What are the domain restrictions for $f(g(x))$?
2. How would you evaluate $g(f(x))$?
3. How do you simplify rational expressions?
4. What happens when $x = 9$ for the function $g(x)$?
5. How does function composition affect the domain and range?

Tip: Always check for any restrictions in the domain when dealing with rational functions, as division by zero can occur.