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Suppose we have say $f(2x)=\left(f(x)\right)^4,$ with $f(0)=1$ and $f$ not being constant. How does one find out what $f$ is without guessing?

More generally, is there a systematic way of finding non-obvious functions that solve $f(x)=g(f(h(x)))?$

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    $f(x)=e^{c x^2}$ can be obtained by assuming $f(x) = e^{(a \hspace{3pt}p(x))}$ and applying $f(0) = 1$.2012-04-07

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