To find the inverse of the function $f(x) = \frac{-2}{x+1} - 2$, follow these steps:

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

$y = \frac{-2}{x+1} - 2$

Step 2: Swap $y$ and $x$

To find the inverse, we switch the roles of $x$ and $y$: $x = \frac{-2}{y+1} - 2$

Step 3: Solve for $y$

Now, we need to solve for $y$ in terms of $x$.

1. First, add 2 to both sides: $x + 2 = \frac{-2}{y+1}$

2. Multiply both sides by $(y+1)$ to eliminate the denominator: $(x + 2)(y + 1) = -2$

3. Expand the left side: $(x + 2)(y + 1) = x(y + 1) + 2(y + 1) = -2$ Simplifying: $(x + 2)y + (x + 2) = -2$

4. Subtract $(x + 2)$ from both sides: $(x + 2)y = -2 - (x + 2)$ Simplifying: $(x + 2)y = -x - 4$

5. Finally, divide by $(x + 2)$: $y = \frac{-x - 4}{x + 2}$

Thus, the inverse function is: $f^{-1}(x) = \frac{-x - 4}{x + 2}$

Would you like more details or have any further questions?

Here are 5 related questions to consider:

1. How can you verify if a function and its inverse are correct?
2. What are the domain and range of the inverse function?
3. How do we find the inverse of a more complex function?
4. What is the graphical relationship between a function and its inverse?
5. How does the method change if the function involves powers or roots?

Tip: To verify if two functions are inverses, compose them: $f(f^{-1}(x))$ and $f^{-1}(f(x))$ should both return $x$.