What is the domain of an inverse function?

Question

If $f:X \to Y$ then if the inverse exists, is the domain the range of $f$ or the codomain of $f$?

Answer 1

I think that it is most easiest to deduce the domain from the definition of inverse function and elementary function theory:

$$ f^{-1}(f(x)) = x $$

Say that $f$ assigns a value $x$ from $X$ to $Y$. To invert that process, $f^{-1}$ must assign a value $y$ from $Y$ to $X$, so in more formal words, if $X$ is the domain of $f$ and the image is $Y$, $Y$ must be the domain of $f^{-1}$

Answer 2

According to Wikipedia the following is common usage today: When we introduce a map $f:\>X\to Y$ then $X$ is the domain, and $Y$ is the codomain of $f$. The set $$f(X):=\{y\in Y\>|\>\exists\>x\in X: \ y=f(x)\}\quad \subset Y$$ of image points is called the image or range of $f$.

If it is assumed or established that $f$ is injective then its inverse map $$f^{-1}:\quad f(X)\to X,\qquad y\mapsto f^{-1}(y),$$ automatically comes into existence. The domain of $f^{-1}$ is clearly the range of $f$, which in general is a proper subset of $Y$.

Note that the name $f^{-1}$ is also used for the preimage of arbitrary subsets $V\subset Y$: $$f^{-1}:\quad {\cal P}(Y)\to{\cal P}(X),\qquad V\mapsto f^{-1}(V):=\{x\in X\>|\> f(x)\in V\}\ .$$ The two uses of $f^{-1}$ are obviously "semantically" related.