Finding the right-inverse of a function

Question

I've been given the following function: $$f:P(X) \to P(X)$$ such that $$ f(A) = X\backslash A$$ I figured out it's surjective and therefore has a right inverse, it's also injective, so it has an inverse.

I am trying to figure out what the right inverse is, I thought about $g(A) = X\backslash A$ but it just gives me an empty set.

Answer 1

Consider $f^2$. For $A ⊆ X$, $$f^2(A) = f(X \setminus A) = X \setminus (X \setminus A) = …$$

What happens if you take everything away from $X$ except for everything in $A$? You’re left with …?