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Question Number 32743 by naka3546 last updated on 01/Apr/18

$$f\left(f\left(n\right)\right)=2n$$$$f\left(n\right)=?$$

Commented by abdo imad last updated on 03/Apr/18

$$iffisapolynomialfof\left(x\right)=2x\Rightarrow f^{,}\left(f\left(x\right)\right)f^{{}^{\prime }}\left(x\right)=2\Rightarrow y^{,}\left(y\right)=\frac{2}{y^{{}^{\prime }}}$$$$degf=p\Rightarrow {degf}^{{}^{\prime }}=p-1\Rightarrow 2(p-1)=0\Rightarrow p=1\Rightarrow f\left(x\right)=\lambda x$$$$f\left(f\left(x\right)\right)=2x\Rightarrow f\left(\lambda x\right)=2x\Rightarrow \lambda \left(\lambda x\right)=2x\mathrm{\forall }x\Rightarrow {\lambda }^2=2\Rightarrow$$$$\lambda =\stackrel{-}{+}\sqrt{2}andf\left(x\right)=\stackrel{-}{+}\sqrt{2}x$$

Answered by Joel578 last updated on 02/Apr/18

$$f\left(n\right)=an$$$$f\left(f\left(n\right)\right)=f\left(an\right)=a\left(an\right)=a^{2}n=2n$$$$a=\pm \sqrt{2}$$$$f\left(n\right)=\pm n\sqrt{2}$$

Answered by MJS last updated on 01/Apr/18

$$f\left(n\right)=\pm n\sqrt{2}$$

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