There are many functions $f$ which satisfy the functional equation $f(f(x))=x$, such as $f(x)=x$, $f(x)=\dfrac1x$, or $f(x)=\sqrt{1-x^2}$.
However, I want a function that satisfies $f(f(x))=e^x$.
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8See https://en.wikipedia.org/wiki/Half-exponential_function – celtschk Apr 30 '17 at 05:53
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See also thoughts about $f(f(x))=e^x$ and Does the exponential function have a square root? for more discussion and links. – dxiv Apr 30 '17 at 06:43
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P.S. Yours is a rather odd-sounding choice for a username. – dxiv Apr 30 '17 at 06:47