The given function is:

$f(x) = x + 4x + 8 = 5x + 8$

We are tasked with finding $f^{-1}(-5)$, which means we need to determine the inverse of $f(x)$ and then substitute $-5$ for $y$ in the inverse.

Step 1: Find the inverse of $f(x)$

To find the inverse, we swap $f(x)$ with $y$, so:

$y = 5x + 8$

Now, solve for $x$ in terms of $y$:

$y - 8 = 5x$

$x = \frac{y - 8}{5}$

Thus, the inverse function is:

$f^{-1}(y) = \frac{y - 8}{5}$

Step 2: Find $f^{-1}(-5)$

Substitute $-5$ for $y$ in the inverse function:

$f^{-1}(-5) = \frac{-5 - 8}{5} = \frac{-13}{5}$

So, $f^{-1}(-5) = -\frac{13}{5}$.

Would you like more details or have any questions?

Here are 5 related questions you could explore:

1. How do you verify if a function has an inverse?
2. What are the steps to find the inverse of a more complex function?
3. How do you determine if a function is one-to-one?
4. What is the graphical interpretation of an inverse function?
5. Can all linear functions have an inverse?

Tip: When finding the inverse of a function, remember to switch the roles of $x$ and $y$ and then solve for $x$.