To find the inverse of the function $f(x) = -5x^3 + 1$, we can proceed with the following steps:

1. Rewrite the function by replacing $f(x)$ with $y$: $y = -5x^3 + 1$

2. Swap $y$ and $x$ to find the inverse: $x = -5y^3 + 1$

3. Solve for $y$:

   • Start by isolating the $y$-term: $x - 1 = -5y^3$
   • Divide by $-5$ to get: $\frac{1 - x}{5} = y^3$
   • Now, take the cube root of both sides to solve for $y$: $y = \sqrt[3]{\frac{1 - x}{5}}$

4. Express the inverse function: $f^{-1}(x) = \sqrt[3]{\frac{1 - x}{5}}$

So, the inverse function is: $f^{-1}(x) = \sqrt[3]{\frac{1 - x}{5}}$

Would you like further details on any step or have any questions?

Here are five related questions to expand your understanding:

1. What is the process to find the inverse of a function in general?
2. Why is it necessary to swap $x$ and $y$ when finding an inverse?
3. Can all functions have inverses? What conditions must a function satisfy to have an inverse?
4. How would the process differ if we were finding the inverse of a quadratic function instead of a cubic function?
5. How can you verify that two functions are inverses of each other?

Tip: When solving for the inverse of a function, make sure to isolate the term involving $y$ before performing operations like square or cube roots.