To find the inverse of the function $y = x^2 - 4$ where $x \leq 0$, we need to follow these steps:

1. Rewrite the function: Start with the given function. $y = x^2 - 4$

2. Express $x$ in terms of $y$: $y + 4 = x^2$ Since $x \leq 0$, we take the negative square root. $x = -\sqrt{y + 4}$

3. Swap $x$ and $y$ to find the inverse function: $y = -\sqrt{x + 4}$

So, the inverse function of $y = x^2 - 4$ for $x \leq 0$ is: $f^{-1}(x) = -\sqrt{x + 4}$

Would you like more details on this process or have any other questions?

Here are 8 related questions that expand on this topic:

1. What is the domain and range of the inverse function $f^{-1}(x) = -\sqrt{x + 4}$?
2. How do you determine the inverse of a function algebraically?
3. What is the graphical relationship between a function and its inverse?
4. How do you verify that two functions are inverses of each other?
5. How would the process change if the domain of the original function was $x \geq 0$?
6. What are some common mistakes to avoid when finding an inverse function?
7. How can you find the inverse of a function if it is not one-to-one?
8. Can all functions have inverses? If not, what are the conditions required for a function to have an inverse?

Tip: Always check the domain and range of the original function and its inverse to ensure they match appropriately.