# What's the name for the property of a function $f$ that means $f(f(x))=x$?

I can think of several examples of functions such that twice application of the function is equivalent to no application of it.

• Multiplicative inverse
• Fourier transform
• Complex conjugation
• Any group built up from $\mathbb{Z}_2$, applying (one of) the $\mathbb{Z}_2$ parts' operation.

"Idempotent" came to mind, but that's wrong. It means $f(f(x)) = f(x)$, not $f(f(x))=x$.

What is the word for this "flip-flop" property?

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Involution – Brandon Carter Nov 26 '11 at 19:12
also known as '$f$ is its own inverse' – Chris Taylor Nov 26 '11 at 19:13
@BrandonCarter Why did you comment instead of answering? I'm not sure whether to "award" the checkmark to you or Leandro. – isomorphismes Nov 26 '11 at 19:41
Fourier transform doesn't have that property ;) You get an extra reflection.... – N. S. Nov 26 '11 at 20:21
possible duplicate of Functions that are their own inversion. – Najib Idrissi Aug 31 '15 at 8:39

Self inverse. I didn't think of that. I googled for something approximately like the title. (the word self inverse didn't come readily to mind because I was thinking $f^2=f^0$ rather than $f^1 = f^{-1}$) – isomorphismes Nov 26 '11 at 21:02