Inverse Compositions and Self-Inverse Functions

Inverse composition identities

If \(\mathrm{f}(x)=y\), then \(\mathrm{f}^{-1}(y)=x\). Thus \(\mathrm{f}^{-1}(\mathrm{f}(x))=x\) and \(\mathrm{f}(\mathrm{f}^{-1}(y))=y\), wherever the compositions are defined.

Self-inverse functions

A self-inverse function, or involution, is its own inverse. Applying it twice returns every permitted input to its starting value.

\[\mathrm{f}(\mathrm{f}(x))=x\qquad\text{or}\qquad\mathrm{f}^{2}(x)=x.\]

Equivalently, \(\mathrm{f}^{-1}=\mathrm{f}\). To prove this, show both \(\mathrm{f}^{-1}(x)=\mathrm{f}(x)\) and \(D_{\mathrm{f}^{-1}}=D_{\mathrm{f}}\).

Repeated composition alternates between the identity and \(\mathrm{f}\):

\[\mathrm{f}^{n}(x)=\begin{cases}x,&n\text{ even},\\\mathrm{f}(x),&n\text{ odd}.\end{cases}\]