We are asked to find the inverse function $f^{-1}(y)$ of the function $f(x) = \frac{1}{4}x - 11$.

1. Problem Description

We are asked to find the inverse function $f^{-1}(y)$ of the function $f(x) = \frac{1}{4}x - 11$.

2. Solution Steps

To find the inverse function, we first replace $f(x)$ with $y$:

$y = \frac{1}{4}x - 11$

Then we switch $x$ and $y$:

$x = \frac{1}{4}y - 11$

Next, we solve for $y$:

$x + 11 = \frac{1}{4}y$

$4(x + 11) = y$

$4x + 44 = y$

So, $y = 4x + 44$.

Finally, we replace $y$ with $f^{-1}(x)$:

$f^{-1}(x) = 4x + 44$

Since we want $f^{-1}(y)$, we replace $x$ with $y$:

$f^{-1}(y) = 4y + 44$

3. Final Answer

$f^{-1}(y) = 4y + 44$