Give an example of sets $A$ and $B$ and functions $F$ and $G$ such that $F: A \rightarrow B, G: B \rightarrow A, G \circ F = I_{A}$, and $G \neq F^{-1}$
I was thinking maybe $F$ can be a function whose inverse does not exist. Say $F(x) = x^{2}$. And $G(x) = \sqrt{x}$, then $G \circ F = x$, but $G \neq F^{-1}$. Does this work out? And $A$ could just be $\left \{1,2 \right \}$ and $B = \left \{1, 4 \right \}$.