The function provided in the image is $f(x) = \frac{1}{x+8}$. To find the inverse function, $f^{-1}(x)$, we need to swap $x$ and $y$ in the function, and solve for $y$:

1. Let $y = f(x) = \frac{1}{x+8}$.
2. Swap $x$ and $y$: $x = \frac{1}{y+8}$.
3. Solve for $y$:

$y + 8 = \frac{1}{x} \quad \Rightarrow \quad y = \frac{1}{x} - 8$

Thus, the inverse function is:

$f^{-1}(x) = \frac{1}{x} - 8$

You can check the correctness of this inverse function by verifying that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Would you like to go through the verification or have any other questions on this topic? Here are a few more related questions:

1. What would be the domain and range of the inverse function?
2. How does the graph of $f(x)$ compare to its inverse?
3. What transformations are involved when finding the inverse of a function?
4. How would this process change if the function included a quadratic term instead of a linear term?
5. Why is the subtraction of 8 important in determining the inverse function?

Tip: Always verify the inverse by substituting back into the original function and checking if the output is $x$.