Some examples of functions that are their own inverse?

Question

I'm looking for the name and some examples of functions $f$ with the following property

$$f\circ f=I$$

where $I$ is the identity. This means that $f=f^{-1}$; some examples are the functions $f(n)=-n$ and $g(n)=1/n$. What are other examples of functions that have this property and what is so special about them?

https://en.wikipedia.org/wiki/Involution_(mathematics) – Oct 31 '17 at 15:55 — , Oct 31 '17 at 15:55
It is called a self-inverse function. – Oct 31 '17 at 16:03 — , Oct 31 '17 at 16:03

score 3 · Accepted Answer · answered Oct 31 '17 at 15:49

3

If, $\forall\, a \in \mathbb{R}$, with $f(x)=a-x$ we find \begin{align} f(f(x)) &= a-(a-x) \\ &= a-a +x \\ &= x \\ \text{i.e.} f^2 &= e \iff f=f^{-1} \end{align}

answered Oct 31 '17 at 15:49

Hector Blandin · Answer 2 · 2017-10-31T15:54:32.860

1

Consider the function $f$ given by $$f(x)=\ln\left(\frac{e^{x}+1}{e^{x}-1}\right)$$

edited Oct 31 '17 at 15:54

answered Oct 31 '17 at 15:50

Hector Blandin

2,471

Oh yes I see :) – Hector Blandin Oct 31 '17 at 15:52

Guy Fsone · Answer 3 · 2017-10-31T16:19:13.513

1

Hint: take $$f(x) =\frac1x~~for ~~~x\neq 0$$

or $$g(x) =\frac{x+1}{x-1}~~~~for ~~x\neq 1.$$

edited Oct 31 '17 at 16:19

answered Oct 31 '17 at 16:11

Guy Fsone

23,903

Some examples of functions that are their own inverse?

3 Answers3