To find the inverse of the function $f(x) = -(x - 1)^3$, follow these steps:

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

$y = -(x - 1)^3$

Step 2: Swap $x$ and $y$ (since we are finding the inverse)

$x = -(y - 1)^3$

Step 3: Solve for $y$ in terms of $x$

1. First, isolate the cubic term by dividing both sides by $-1$: $-x = (y - 1)^3$

2. Take the cube root of both sides to undo the cubic: $\sqrt[3]{-x} = y - 1$

3. Finally, solve for $y$ by adding 1 to both sides: $y = \sqrt[3]{-x} + 1$

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

Thus, the inverse function is: $f^{-1}(x) = \sqrt[3]{-x} + 1$

Final Answer:

$f^{-1}(x) = \sqrt[3]{-x} + 1$

Would you like to dive deeper into how this process works or have any other questions? Here's a list of related topics:

1. How do we determine if a function has an inverse?
2. What does the graph of a function and its inverse look like?
3. How do we check if two functions are inverses of each other?
4. What are the properties of inverse functions?
5. How do cube roots and other radical functions behave?

Tip: For inverse functions, always remember that swapping $x$ and $y$ is the key first step in finding the inverse.