Does $f(f(x))=x$ imply surjectivity?

Question

If for $f: \mathbb{R} \rightarrow \mathbb{R}$ $f(f(x))=x \quad \forall x \in \mathbb{R}$, can we say that the function is surjective?

I came across this problem while solving a different problem, and I think the answer should be yes, because all such real $x$ is getting mapped to by the function. So it should be surjective, however I'm not sure as I'm quite new in learning functions, so I want to verify my answer. Thank you.

for further reading, google the terms left inverse and right inverse — C Squared, Apr 14 '23 at 18:54

score 4 · Accepted Answer · answered Apr 14 '23 at 17:54

4

Yes, we can, for the reason you mention.

By definition, $f$ is surjective iff:

For all $x\in\Bbb R$ there exists $y\in\Bbb R$ with $f(y)=x$

Since for any given $x$, you can take $y:=f(x)$ and have $f(y)=f(f(x))=x$, $f$ is surjective.

Such $f$ are called involutions.

There is absolutely nothing special about $\Bbb R$ here. If $X$ is any set, $f:X\to X$ any involution (that is, $f(f(x))=x$ for all $x\in X$) then it follows $f$ is surjective.

answered Apr 14 '23 at 17:54

FShrike

40,125

1

For the OP: involutions are even bijective. For FShrike: it was a duplicate. – Anne Bauval Apr 14 '23 at 17:57

Does $f(f(x))=x$ imply surjectivity?

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