inverse function theorem

An inverse function \( f^{-1} \) exists for a function \( f : X \to Y \) if and only if \( f \) is bijective. The inverse \( f^{-1} : Y \to X \) satisfies \( f^{-1}(f(x)) = x \) for all \( x \in X \) and \( f(f^{-1}(y)) = y \) for all \( y \in Y \).