To find the inverse of the function $f(x) = \frac{1}{x} + 15$, follow these steps:

1. Replace $f(x)$ with $y$: $y = \frac{1}{x} + 15$

2. Swap $x$ and $y$ (since we want to solve for the inverse): $x = \frac{1}{y} + 15$

3. Solve for $y$:

   • Subtract 15 from both sides: $x - 15 = \frac{1}{y}$
   • Take the reciprocal of both sides to isolate $y$: $y = \frac{1}{x - 15}$

Thus, the inverse function is: $f^{-1}(x) = \frac{1}{x - 15}$

Would you like further details or explanations?

Here are 5 related questions to expand your understanding:

1. How do you verify that two functions are inverses of each other?
2. What is the domain of the inverse function $f^{-1}(x) = \frac{1}{x - 15}$?
3. What happens to the inverse function if the original function has an additional transformation?
4. How would you graph both $f(x)$ and $f^{-1}(x)$?
5. Can an inverse function always exist for any given function?

Tip: Always check the domain and range of the function and its inverse, as not all functions are invertible over their entire domain.