Name for mapping inputs leads to same output of a function?

Question

For a two-to-one function, $f(i) = f(j) = k$, I want to get a new function/mapping $g(i) = j$, such that f(g(x)) = f(x).

Is there a name for getting such $g$ given $f$ ?

If helpful, $f(x) = |x -1|$, then i have $g(x) = 2-x$.