To find the inverse of the function $f(x) = \frac{1}{x} + 13$, we follow these steps:

1. Replace $f(x)$ with $y$: $y = \frac{1}{x} + 13$

2. Swap $y$ and $x$ to set up the equation for the inverse function: $x = \frac{1}{y} + 13$

3. Isolate $y$ by subtracting 13 from both sides: $x - 13 = \frac{1}{y}$

4. Take the reciprocal of both sides to solve for $y$: $y = \frac{1}{x - 13}$

Thus, the inverse function is: $f^{-1}(x) = \frac{1}{x - 13}$

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you verify if two functions are inverses of each other?
2. What is the domain of $f^{-1}(x)$?
3. How does the graph of $f(x)$ compare to the graph of $f^{-1}(x)$?
4. How do inverse functions affect compositions, like $f(f^{-1}(x))$?
5. How does the transformation $+13$ in the original function affect the inverse?

Tip: When finding the inverse, always make sure to check the domain of both the original and inverse functions to avoid undefined values.