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

  • Additive inverse
  • 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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    @AlexM. You're the one being pedant about definitions. The time two questions were asked is irrelevant when $c$onsidering which to close as a duplicate of the other. Only the usefulness of $b$oth is. Why do you think it's possi$b$le to close $a$n older question as a dupli$c$ate of a newer one? It's not an oversight. If you're unhappy, open a meta thread.2015-08-31

2 Answers 2

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I think this is called an "Involution".

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    this is the accepted answer. therefore, please elaborate more and/or cite your sources.2016-12-31
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Involution is the most common name. They are so fundamental that an entire book has been written on them, the Book of Involutions. I often emphasize their essential role both here and various other places. One should always strive to bring to the fore the innate symmetries in problems, and involutions are one of the simplest examples.

Note: you could have found the answer simply by Googling "self inverse function". The first match is the "self inverse" section of the Wikipedia page on inverse functions, which states "such a function is called an involution".

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    @Lao It is well-worth the effort to learn how to compose effective searches, by means of which you can truly **stand on the shoulder of giants** - exposing many beautiful mathematical vistas.2011-11-26