To find the inverse of a function, we reverse the roles of $x$ and $y$, and then solve for $y$.

Given function:

$f(x) = \frac{1}{x}$

Steps to find the inverse:

1. Start by writing the function as $y = \frac{1}{x}$.
2. Switch $x$ and $y$ to reflect the inverse: $x = \frac{1}{y}$
3. Solve for $y$: $y = \frac{1}{x}$

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

$f^{-1}(x) = \frac{1}{x}$

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. What is the inverse of the function $f(x) = 2x + 3$?
2. How can you verify if a function and its inverse are correct?
3. Is the function $f(x) = \frac{1}{x}$ one-to-one and onto?
4. What is the domain and range of $f(x) = \frac{1}{x}$ and its inverse?
5. How would you find the inverse of $f(x) = \frac{1}{x + 2}$?

Tip: A function must be one-to-one (bijective) for it to have an inverse. This means it passes both the horizontal line test and is onto.