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This is a problem that a friend of mine gave to me a while back. I somewhat solved it and paid no attention to it afterwards until I though about it now. Forgive the errors and laziness this was a long time ago and not everything has to be perfect.

I assumed $f$ to be a polynomial function of the kind $f(x)=ax^b$ where $f^{-1}(x)= \big(\frac x a\big)^{1/b} $ and found $$f(x)= \varphi^{-\frac{1}{\varphi}-1}(-x)^{\phi} $$

where $\varphi=\frac{1+\sqrt{5}}{2}$ and $\phi=\frac{1-\sqrt{5}}{2}$ my working is in the following images : (I did this in word)

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I was curious if there were more solutions to just this polynomial? Perhaps a different function ?

Don't bother commenting trying to correct my work I am just curious about the above questions. Thank you for your time

hwood87
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    Relevant MO thread: https://mathoverflow.net/questions/34052/function-satisfying-f-1-f – lc2r43 Sep 03 '21 at 04:20
  • @lc2r43 Thank you – hwood87 Sep 03 '21 at 04:21
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    Previous discussion https://math.stackexchange.com/questions/735695/function-whose-inverse-is-also-its-derivative, other same questions https://math.stackexchange.com/questions/1660413/when-could-we-get-f-f-1-where-f-1-is-the-inverse-function-of-f, https://math.stackexchange.com/questions/1916191/ode-with-inverse-function-solve-f-1x-fx – Lutz Lehmann Sep 03 '21 at 04:42
  • Michael Penn takes a look at it: https://www.youtube.com/watch?v=rNUfiQgj6ZI – md2perpe Sep 03 '21 at 06:33

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