The given function is $f(x) = |x| - 1$, for $x \leq 0$, and the task is to find its inverse $f^{-1}(x)$.

Steps to find the inverse:

1. Rewrite the function: The absolute value condition $|x|$ for $x \leq 0$ simplifies to $f(x) = -x - 1$. So the function becomes: $f(x) = -x - 1$ for $x \leq 0$.

2. Swap $x$ and $y$: To find the inverse, swap $f(x)$ with $y$ and then swap $x$ and $y$: $x = -y - 1$

3. Solve for $y$: $y = -x - 1$ Thus, the inverse function is $f^{-1}(x) = -x - 1$.

Domain and Range:

• The domain of the original function $f(x)$ is $(-\infty, 0]$.
• The range of the original function is $(-\infty, -1]$ (since $|x| - 1 \leq -1$).

For the inverse $f^{-1}(x)$, the domain and range swap:

• Domain: $(-\infty, -1]$.
• Range: $(-\infty, 0]$.

Therefore, the equation for the inverse is:

$f^{-1}(x) = -x - 1$ for $x \leq -1$.

Do you want more details or have any questions?

Here are five related questions:

1. How do you determine the domain and range of a function and its inverse?
2. What happens when you apply the inverse function to the original function?
3. Can an inverse function exist if the original function is not one-to-one?
4. How do absolute value functions affect the calculation of an inverse?
5. Why is it important to restrict the domain when finding an inverse function?

Tip: Always check that the inverse function satisfies both the original function's domain and range when verifying your answer!