Inverse relations notation (not a function)

Question

Even if $f:X\to Y$ is not a bijective function, can I still notate the inverse relation of $f$ as $g:Y\to X$?

Answer 1

Normally the notation $g:Y\to X$ is used to mean that $g$ is a function from $Y$ to $X$ (though it can have other meanings in certain contexts). So by writing $g:Y\to X$, you would be asserting that $g$ is a function with domain $Y$, which is true only when $f$ is a bijection. The notation $g:Y\looparrowright X$ is sometimes used to denote "$g$ is a relation from $Y$ to $X$".