Find monotone functions $f:\Bbb{R}^\to\Bbb{R}^$ such that $f\left (\frac{1}{f\left(\frac{1}{f(x)}\right)}\right)=x,\forall x\in\Bbb R^*$

Asked Mar 14 '20 at 22:25

Active Mar 14 '20 at 23:45

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Find all monotone functions $f: \mathbb{R^{*}} \rightarrow \mathbb{R^{*}}$ with this propriety $$f\left ( \frac{1}{f\left ( \frac{1}{f(x)} \right )} \right )=x,\forall x\in \mathbb R^{*}$$

Any help would be appreciated. I don't really know how to start this problem, I haven't found anything similar with it.(it's a 9th grade problem)

edited Mar 14 '20 at 23:45

Shaun

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asked Mar 14 '20 at 22:25

omega123

1

have u tried f(x)=x ? – zeraoulia rafik Mar 14 '20 at 22:37
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One way to start this problem is to try different forms of $ax^b$, $a^x$, $\log x$, and then seeing how the end up, it might get close to $x$ and you can adjust it from there. Obviously if you have for instance: $f(f(f(x)))=x$, the solution will never be $a^x$ since you'll just get $a^{a^{a^{x}}}$, and you can look to the next attempt. For this example, I would personally start with $f(x)=\frac1x$ and see if that works. – Jacob Claassen Mar 14 '20 at 22:49
Ok, thank you, I will try – omega123 Mar 14 '20 at 22:52
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For simplicity, you want to let $g(x)=f(1/x)$. This boils down to a problem analogous to f(f(x)) = 1/x. – Simply Beautiful Art Mar 14 '20 at 22:53
Thank you for the idea! – omega123 Mar 14 '20 at 22:55
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As a matter of fact, by working out the above further by applying the entire resulting equation to itself in the same manner as shown in the link, one gets this. – Simply Beautiful Art Mar 14 '20 at 22:57

Find monotone functions $f:\Bbb{R}^*\to\Bbb{R}^*$ such that $f\left (\frac{1}{f\left(\frac{1}{f(x)}\right)}\right)=x,\forall x\in\Bbb R^*$

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Find monotone functions $f:\Bbb{R}^\to\Bbb{R}^$ such that $f\left (\frac{1}{f\left(\frac{1}{f(x)}\right)}\right)=x,\forall x\in\Bbb R^*$