Restricted domains and self-inverse functions

A many-to-one rule may become one-to-one after restricting its domain. The chosen branch determines the inverse. Example For f(x)=(x-1)2, choose x 1. Then f-1(x)=1+sqrt(x), x 0. Choosing x 1 instead gives f-1(x)=1-sqrt(x). A self-inverse function satisfies f(f(x))=x for every input and maps its domain onto itself. For f(x)=5-x, f(f(x))=5-(5-x)=x on R.