# 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?

• Involution
– bzc
Commented Nov 26, 2011 at 19:12
• also known as '$f$ is its own inverse' Commented Nov 26, 2011 at 19:13
• @BrandonCarter Why did you comment instead of answering? I'm not sure whether to "award" the checkmark to you or Leandro. Commented Nov 26, 2011 at 19:41
• Fourier transform doesn't have that property ;) You get an extra reflection.... Commented Nov 26, 2011 at 20:21
• possible duplicate of Functions that are their own inversion. Commented Aug 31, 2015 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 $f^1 = f^{-1}$) Commented Nov 26, 2011 at 21:02