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How should i solve expressions in $f$?

$$ f(f(x))=e^x $$

or:

$$ f(f(x+1)+1)=x $$

Series, expansions, discretizations, CAS, whatever?

My normal approach is useless here...

Brethlosze
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  • What is your normal approach? – projectilemotion May 28 '17 at 22:52
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    https://xkcd.com/55/ – Brethlosze May 28 '17 at 22:53
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    This is way to broadly formulated and with way to little context. Do you mean any equation or just these two? – Winther May 28 '17 at 22:53
  • I mean, those look always as simple expressions, but most of them dont show a simple solution at all..... – Brethlosze May 28 '17 at 22:54
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    Take a look at these related questions: [thoughts about $f(f(x))=e^x$](https://math.stackexchange.com/questions/65876/thoughts-about-ffx-ex) ; [How to calculate $f(x)$ in $f(f(x))=e^x$?](https://math.stackexchange.com/questions/59023/how-to-calculate-fx-in-ffx-ex) ; [Find all continuous functions $f:\mathbb R\to\mathbb R$ such that $f(f(x))=e^{x}$](https://math.stackexchange.com/questions/1087833/find-all-continuous-functions-f-mathbb-r-to-mathbb-r-such-that-ffx-ex) ; [Find all entire $f$ such that $f(f(z))=z$](https://math.stackexchange.com/questions/623457/find-all-entire-f-such-that-ffz-z) – Winther May 28 '17 at 22:55
  • the second $f(f(x+1)+1)=x$ is $f(x)=x-1$ – Jacob Claassen May 28 '17 at 22:57
  • I never though that would be so difficult and complex....... – Brethlosze May 28 '17 at 22:59
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    See [this wikipedia article](https://en.wikipedia.org/wiki/Half-exponential_function). – Arthur May 28 '17 at 23:09
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    Assume $f(x)=ax+b$ for the second one. – Simply Beautiful Art May 28 '17 at 23:11

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I don't know how to solve the first one, but just by guessing, one solution to the it is $f(x) = x-1$, so $f(x+1) = x$, $f(x+1)+1 = x+1$, $f(f(x+1)+1) = x$

Yimo Awanardo
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