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Question Number 112681 by bemath last updated on 09/Sep/20

$$\mathrm{solve}\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{function}$$$${\left(\mathrm{f}\left(3\mathrm{x}\right)\right)}^2={\left(\mathrm{f}\left(2\mathrm{x}\right)\right)}^2+{\left(\mathrm{f}\left(\mathrm{x}\right)\right)}^2$$

Answered by bobhans last updated on 09/Sep/20

$$\left(⧫\right){\left(\mathrm{f}\left(3\mathrm{x}\right)\right)}^2={\left(\mathrm{f}\left(2\mathrm{x}\right)\right)}^2+{\left(\mathrm{f}\left(\mathrm{x}\right)\right)}^2$$$$\mathrm{by}\mathrm{letting}\mathrm{h}\left(\mathrm{x}\right)={\left(\mathrm{f}\left(\mathrm{x}\right)\right)}^2$$$$\mathrm{now}\mathrm{we}\mathrm{get}\mathrm{h}\left(3\mathrm{x}\right)=\mathrm{h}\left(2\mathrm{x}\right)+\mathrm{h}\left(\mathrm{x}\right)$$$$\mathrm{plugging}\mathrm{in}\mathrm{x}=0$$$$\Rightarrow \mathrm{h}\left(0\right)=2\mathrm{h}\left(0\right),\Rightarrow \mathrm{h}\left(0\right)=0$$$$\mathrm{since}\mathrm{the}\mathrm{equation}\mathrm{is}\mathrm{linear},\mathrm{let}\mathrm{h}\left(\mathrm{x}\right)=\mathrm{Cx}$$$$\mathrm{therefore}\mathrm{f}\left(\mathrm{x}\right)=\pm \sqrt{\mathrm{Cx}}\mathrm{or}\mathrm{f}\left(\mathrm{x}\right)=\mathrm{A}\sqrt{\mathrm{x}}$$$$\mathrm{where}\mathrm{A}\mathrm{can}\mathrm{be}\mathrm{any}\mathrm{constant}$$

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