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Question Number 57011 by 121194 last updated on 28/Mar/19

$$f\left(\frac{x+y}{2}\right)f\left(\frac{x-y}{2}\right)=g\left(x\right)$$$$g(x+y)g(x-y)={\left[f\left(x\right)\right]}^2-{\left[f\left(y\right)\right]}^2$$$$f\left(x\right),g\left(x\right)=?$$

Answered by kaivan.ahmadi last updated on 28/Mar/19

$$g\left(x\right)$$$$★x=y=0\Rightarrow f\left(0\right)f\left(0\right)=g\left(0\right)$$$$g\left(0\right)g\left(0\right)=0\to g\left(0\right)=0\to f\left(0\right)=0$$$$★x=y\Rightarrow$$$$f\left(x\right)f\left(0\right)=g\left(x\right)\Rightarrow g\left(x\right)=0$$$$★y=0\Rightarrow$$$$g\left(x\right)g\left(x\right)={\left[f\left(x\right)\right]}^2\Rightarrow {\left[f\left(x\right)\right]}^2=0\Rightarrow f\left(x\right)=0$$$$$$$$$$

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