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Let $y$ be a real number.

Find $g$ such that

$$g(x + \frac{1}{x}) = x^y + \frac{1}{x^y}$$

Is valid for all real $x$.

mick
  • 17,886
  • What did you try, what are your thoughts? $x+1/x=u$ is a quadratic equation in $x$, it should be trivial to find a form for $g(u)$. – Lutz Lehmann Dec 17 '15 at 12:35
  • Why close and downvote ? – mick Dec 17 '15 at 20:05
  • Because your question is incomplete. You did not show your own efforts until the breaking point that you could not overcome. Usually, this place is not meant to solve your homework or arbitrary riddles, but to help with your specific problems in solving them. – Lutz Lehmann Dec 17 '15 at 20:44
  • It was not homework. – mick Dec 17 '15 at 20:46
  • Related : http://math.stackexchange.com/questions/1578246/about-fx-f-frac1x – mick Dec 17 '15 at 20:49

1 Answers1

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Setting $x=e^u$ one gets $$g(2\cosh(u))=2\cosh(u·y)$$ which resolves to $$ g(z)=2\cosh(y·\text{Arcosh}(z/2)) $$ for all $z\ge 2$

Lutz Lehmann
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