Self-inverse functions

Definition

A special type of function is the self-inverse function: a function that is its own inverse. That is, $f(x)=f^{-1}(x)$ for all $x$ in the domain of $f.$

For example, we can check that $f(x)=x,$ $g(x)=-x$ and $h(x)=\frac{1}{x}$ are all self-inverse functions.

Special results for self-inverse functions

In the previous sections saw that $f^2(x)=ff(x)$ and $f{f}^{-1}(x)=x.$ For self inverse functions, since $f=f^{-1},$ we have $f^2(x)=ff(x)=f{f}^{-1}(x)=x.$

The result that $f^2(x)=x$ is rather powerful: it means that applying $f$ twice has the same result of not doing anything. This allows us to simplify cases where we compose a function several times.

For example, $f^3(x)=f{f}^2(x)=f(x)$ and $f^4(x)=f^2{f}^2(x)=f^2(x)=x.$ Are you able to figure out what $f^{314}(x)$ and $f^{999}(x)$ are?