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This question already has an answer here:

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?

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marked as duplicate by user7530, drhab, Guy Fsone, Community Oct 31 '17 at 20:14

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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}

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

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

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

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  • $\begingroup$ Oh yes I see :) $\endgroup$ – Hector Blandin Oct 31 '17 at 15:52

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