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I have to find all functions so that

$$ f\left(\frac{x}{f(y)}\right) = \frac{x}{f(x\sqrt{y})} $$

I have no idea how to solve this one. Any help would be appreciated!

Victor
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  • If you just need an example, try $f(x)=\sqrt{x}$. – Redundant Aunt Feb 03 '16 at 22:00
  • There is a big difference between "find one" and "find all". Is the question as written the one you want to ask (given your comments under the answer it seems otherwise)? – Winther Feb 03 '16 at 22:14
  • Moreover, it is not clear what should be the domain of the function (and the domain of the functional equation). I mean, if we are looking for $f:\mathbb{R} \to \mathbb{R}$ then $f(x)=\sqrt{x}$ is surely not a solution.. – Tintarn Feb 03 '16 at 22:42
  • The edit changes the problem. In my opinion, it makes it considerably harder to solve; at least, it is apparent that it invalidates the existing aswers. Yet, the mathematical effort towards the solution remains the same (zero). Therefore, I see no reason to vote to reopen it. –  Feb 04 '16 at 05:47

2 Answers2

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Hint: try a power function $f(x)=x^{p}$ and see what $p$ would have to be.

2'5 9'2
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  • This seems more like an intuitive solution, doesn't it? Don't I have to prove anything about this function? (I tend to say: Shouldn't I prove that the function is unique? but that may not be true) – Victor Feb 03 '16 at 22:10
  • Your question says "find a function $f$ so that..." which means that if you can find even one function (like $x^{1/2}$ which results from this investigation) then you have replied to the question. As posted, your question says nothing about finding "the only" function with the given property, so there is no burden to prove uniqueness (if that's even true). – 2'5 9'2 Feb 03 '16 at 22:30
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$f(x)=\sqrt x$ is a solution of your problem

stity
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