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Question Number 146017 by Khalmohmmad last updated on 10/Jul/21

Answered by Olaf_Thorendsen last updated on 10/Jul/21

$$F\left(x\right)+F\left(y\right)=F\left(x\right)F\left(y\right)\left(1\right)$$$$x=y\Rightarrow 2F\left(x\right)=F^2\left(x\right)\iff F\left(x\right)(F\left(x\right)-2)=0$$$$\mathrm{\forall }x,F\left(x\right)=0\mathrm{or}2:\mathrm{impossible}$$$$\mathrm{because}\mathrm{F}\left(\frac{1}{2}\right)=\frac{3}{2}$$$$\Rightarrow \mathrm{incompatible}\mathrm{conditions}!$$$$\mathrm{We}\mathrm{verifly}\mathrm{that}\mathrm{if}\mathrm{F}\left(\frac{1}{2}\right)=\frac{3}{2}:$$$$\left(1\right)\mathrm{F}\left(\frac{1}{2}\right)+\mathrm{F}\left(\frac{1}{2}\right)(=3)\ne \mathrm{F}\left(\frac{1}{2}\right)\mathrm{F}\left(\frac{1}{2}\right)(=\frac{9}{4})$$

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