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

Step 1: Write the function as $y = f(x)$

$y = \frac{1}{x} + 15$

Step 2: Solve for $x$ in terms of $y$

First, subtract 15 from both sides to isolate the term involving $x$: $y - 15 = \frac{1}{x}$ Now, take the reciprocal of both sides to solve for $x$: $x = \frac{1}{y - 15}$

Step 3: Replace $y$ with $x$ to find $f^{-1}(x)$

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

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

Would you like more details or clarifications?

Here are 5 related questions to explore:

1. How do we verify that two functions are inverses?
2. What happens to the domain and range when finding inverse functions?
3. How can inverse functions be applied to real-world problems?
4. What are the limitations when finding the inverse of a function?
5. How does the graph of a function compare to its inverse?

Tip: When checking if two functions are inverses, remember that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.