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Square root of a function (in the sense of composition)

I'm interested in solving equations of the form $f(f(x))=g(x)$ for $x\in\mathbb{R}$ where $g(x)$ is a known function.

For example: if $g(x)=x$ then we $f(f(x))=x$ so one solution is $f(x)=x$, $\forall x\in\mathbb{R}$. However, another is

$f(x)=x+1$ if $x\in(0,1]$, $f(x)=x-1$ if $x\in(1,2]$ and $f(x)=x$ otherwise.

Hence there are infinitely many solutions to this equation when $g(x)=x$.

Problem: Find an $f$ where $f(f(x))=e^x$.

Extra: Are there extra constraints that could be placed on $f$ so the solution is unique?

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alext87
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    To what extent is this is a duplicate of http://math.stackexchange.com/questions/3633/square-root-of-a-function-in-the-sense-of-composition ? – Isaac Sep 19 '10 at 16:38
  • I apologise for the similarity, that post was posted before I knew about this site and the search didn't bring up anything. – alext87 Sep 19 '10 at 16:56
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    Also a duplicate of http://math.stackexchange.com/questions/1118/characterising-functions-f-that-can-be-written-as-f-g-g/1122# . – Qiaochu Yuan Sep 19 '10 at 17:27
  • Perhaps it is best if this question is closed. I apologise for the inconvenience. – alext87 Sep 19 '10 at 17:32
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    Were you able to solve this from the other links? –  Sep 19 '10 at 19:55
  • The links provided here were also linked to the math overflow site where I have found the answers I was looking for. – alext87 Sep 19 '10 at 22:11

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