Inverse branch ($f\circ g=\mathrm{id}$)

An inverse branch of a function $f$ on a set $J$ is an inverse function $g:J\to I$ of a bijective restriction $f{|}_I:I\to J$. Thus $f(g(y))=y$ for every $y\in J$, and $g(f(x))=x$ for every $x\in I$.

Different restrictions can give different branches on the same $J$. For $f(x)=x^2$, the restrictions to $[0,\infty )$ and $(-\infty ,0]$ give $g_{+}(y)=\sqrt{y}$ and $g_{-}(y)=-\sqrt{y}$ on $J=[0,\infty )$. Their function compositions describe successive choices of preimages, as in the itinerary cylinders of an interval map.