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Question Number 106023 by ZiYangLee last updated on 02/Aug/20

$$\mathrm{Given}\mathrm{that}\mathrm{f}\left(0\right)\ne 0\mathrm{for}\mathrm{all}\mathrm{x},\mathrm{y}\in \mathbb{R};$$$$2\mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{y}\right)=\mathrm{f}(\mathrm{x}+\mathrm{y})+\mathrm{f}(\mathrm{x}-\mathrm{y}),$$$$\mathrm{express}\mathrm{f}\left(2\mathrm{x}\right)\mathrm{in}\mathrm{term}\mathrm{of}\mathrm{f}\left(\mathrm{x}\right).$$

Commented by mr W last updated on 02/Aug/20

$$y=0:$$$$2f\left(x\right)f\left(0\right)=f\left(x\right)+f\left(x\right)$$$$\Rightarrow f\left(0\right)=1$$$$y=x:$$$$2f\left(x\right)f\left(x\right)=f\left(2x\right)+f\left(0\right)$$$$\Rightarrow f\left(2x\right)=2{\left(f\left(x\right)\right)}^2-1$$

Commented by ZiYangLee last updated on 02/Aug/20

$$$$$$\mathrm{Wow}\mathrm{Nice}\mathrm{Thanks}\mathrm{for}\mathrm{your}\mathrm{solutions}$$

Answered by Aziztisffola last updated on 02/Aug/20

$$2\mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(2\mathrm{x}\right)+\mathrm{f}\left(0\right)$$$$\mathrm{f}\left(2\mathrm{x}\right)={2\mathrm{f}}^2\left(\mathrm{x}\right)-\mathrm{f}\left(0\right)$$

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