To find the inverse of the function $f(x) = \frac{1}{x} + 4$, we need to follow these steps:

1. Replace $f(x)$ with $y$: $y = \frac{1}{x} + 4$

2. Swap $x$ and $y$ to solve for the inverse: $x = \frac{1}{y} + 4$

3. Solve for $y$: $x - 4 = \frac{1}{y}$ $y = \frac{1}{x - 4}$

So, the inverse function is: $f^{-1}(x) = \frac{1}{x - 4}$

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:

1. How do you verify that $f(f^{-1}(x)) = x$?
2. What is the domain and range of $f^{-1}(x) = \frac{1}{x - 4}$?
3. How do you find the inverse of a more complicated rational function?
4. Can all functions have an inverse? If not, why?
5. How do transformations (like shifting or scaling) affect the inverse of a function?

Tip: The domain of an inverse function is the range of the original function, and vice versa. Keep this in mind when solving inverse problems!