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Question Number 2000 by Yozzi last updated on 29/Oct/15

$$Findanon-constantfunctionf$$$$satisfyingf\left(0\right)=1,f(-2)=0and$$$$f(x-y)=f\left(x\right)f\left(y\right)-f(-2-x)f(y-2).$$

Commented by prakash jain last updated on 29/Oct/15

$$x=0$$$$f(-y)=f\left(0\right)f\left(y\right)-f(-2)f(y-2)$$$$\because f(-2)=0\mathrm{for}\mathrm{the}\mathrm{above}\mathrm{relation}\mathrm{to}\mathrm{be}$$$$\mathrm{true}\mathrm{\forall }y.$$$$f(-y)=f\left(y\right)$$$$\Rightarrow evenfunction$$$$y=x$$$$f\left(0\right)={\left[f\left(x\right)\right]}^2-f(-2-x)f(x-2)$$$${\left[f\left(x\right)\right]}^2=1+f(x-2)f(x+2)$$

Commented by prakash jain last updated on 30/Oct/15

$$y=2$$$$f(x-2)=f\left(x\right)f\left(2\right)-f(x+2)f\left(0\right)$$$$f(x-2)=-f(x+2)$$$$f\left(x\right)=-f(x+4)$$$$f(x+8)=\left[f(x+4+4)\right]=-f(x+4)=f\left(x\right)$$$$fisperiodicwithperiod8.$$$$\cos \left[\frac{\pi }{4}(x-y)\right]=\cos \frac{\pi }{4}x\cdot \cos \frac{\pi }{4}y+\sin \left(\frac{\pi }{4}x\right)\sin \left(\frac{\pi }{4}y\right)$$$$\sin \frac{\pi y}{4}=\cos \left(\frac{\pi (y-2)}{4}\right)$$$$\sin \frac{\pi x}{4}=-\cos (\frac{\pi }{2}+\frac{\pi }{4}x)=-\cos \left(\frac{\pi (-x-2)}{4}\right)$$

Answered by Rasheed Soomro last updated on 02/Nov/15

$$Findanon-constantfunctionf$$$$satisfyingf\left(0\right)=1,f(-2)=0and$$$$f(x-y)=f\left(x\right)f\left(y\right)-f(-2-x)f(y-2)...........\left(1\right)$$$$------\times \times \times -------$$$$x=-2,y=x$$$$f(-2-x)=f(-2)f\left(x\right)-f(-2-\{-2\})f(x-2)$$$$f(-2-x)=0.f\left(x\right)-f\left(0\right)f(x-2)=-f(x-2)$$$$f(-2-x)=-f(x-2)=-\{f\left(x\right)f\left(2\right)-f(-2-x)f(2-2)\}$$$$=-\{f\left(x\right)f\left(2\right)-f(-2-x)f\left(0\right)\}=f(-x-2)-f\left(2\right)f\left(x\right)$$$$f(-x-2)-f(-x-2)=-f\left(2\right)f\left(x\right)$$$$f\left(2\right)=0\mid f\left(x\right)=0[Butthisnotnon-constant]$$$$\ast \ast \ast$$$$Substitutingy=0$$$$f\left(x\right)=f\left(x\right)f\left(0\right)-f(-2-x)f(-2)$$$$=f\left(x\right)$$$$Thismeansf\left(x\right)isarbitraryify=0.$$$$x=-2-x$$$$f(-2-x-y)=f(-2-x)f\left(y\right)-f(-2-\left\{\right\})$$$$continue$$

Answered by prakash jain last updated on 30/Oct/15

$$f\left(x\right)=\cos \frac{\pi x}{4}$$

Commented by prakash jain last updated on 30/Oct/15

$$seecommentsinquestionforexplaination.$$

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