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.

A many-to-one rule may become one-to-one after restricting its domain. The chosen branch determines the inverse.

For \(\mathrm{f}(x)=(x-1)^2\), choose \(x\ge1\). Then \(\mathrm{f}^{-1}(x)=1+\sqrt{x}\), \(x\ge0\). Choosing \(x\le1\) instead gives \(\mathrm{f}^{-1}(x)=1-\sqrt{x}\).

A self-inverse function satisfies \(\mathrm{f}(\mathrm{f}(x))=x\) for every input and maps its domain onto itself. For \(\mathrm{f}(x)=5-x\), \(\mathrm{f}(\mathrm{f}(x))=5-(5-x)=x\) on \(\mathbb R\).

Functions

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Restricted domains and self-inverse functions

Restricted domains and self-inverse functions