Given the function $f(x) = \frac{4}{-7x-8}$, we need to find its inverse function $f^{-1}(y)$.

1. Problem Description

Given the function $f(x) = \frac{4}{-7x-8}$, we need to find its inverse function $f^{-1}(y)$.

2. Solution Steps

To find the inverse function, we follow these steps:

1. Replace $f(x)$ with $y$:

$y = \frac{4}{-7x-8}$

2. Swap $x$ and $y$:

$x = \frac{4}{-7y-8}$

3. Solve for $y$:

$x(-7y-8) = 4$

$-7xy - 8x = 4$

$-7xy = 4 + 8x$

$y = \frac{4+8x}{-7x}$

$y = \frac{-4-8x}{7x}$

We can also write it as $y = -\frac{4+8x}{7x}$.

4. Replace $y$ with $f^{-1}(x)$:

$f^{-1}(x) = \frac{4+8x}{-7x}$

We are looking for $f^{-1}(y)$, so we just replace $x$ with $y$:

$f^{-1}(y) = \frac{4+8y}{-7y}$

$f^{-1}(y) = \frac{-(4+8y)}{7y}$

$f^{-1}(y) = \frac{-4-8y}{7y}$

We can also derive it like this:

$-7xy - 8x = 4$

$-7xy = 4 + 8x$

$y = \frac{4 + 8x}{-7x}$

However, we want to express in term of $y$, so

$x = \frac{4}{-7y - 8}$

$x(-7y-8) = 4$

$-7xy - 8x = 4$

$-7xy = 8x + 4$

$y = \frac{8x+4}{-7x}$

If the question requires finding $f^{-1}(y)$, we replace $x$ by $y$, then

$-7yx - 8x = 4$

$-7yx = 4+8x$

$-7y = \frac{4+8x}{x}$

$-7xy-8x = 4$

$-7xy = 4+8x$

$x = \frac{4}{-7y-8}$

$-7xy-8x = 4$

$-7xy = 4+8x$

$x = \frac{4}{-7y-8}$

$-7xy - 8x = 4$

$-7xy = 4 + 8x$

Swap x and y:

$-7yx - 8y = 4$

$-7yx = 4+8y$

$x = \frac{4+8y}{-7y}$

3. Final Answer

$f^{-1}(y) = \frac{4+8y}{-7y}$

The answer is $f^{-1}(y) = \frac{4+8y}{-7y}$ which can also be written as $f^{-1}(y) = \frac{-4-8y}{7y}$.