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f-R-R-g-R-R-f-x-y-f-x-f-y-g-y-g-x-y-f-x-g-x-g-y-

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Question Number 650 by 123456 last updated on 19/Feb/15
f:R→R g:R→R f(x+y)=f(x)+f(y)g(y) g(x+y)=f(x)g(x)+g(y)
f:R→Rg:R→Rf(x+y)=f(x)+f(y)g(y)g(x+y)=f(x)g(x)+g(y)
Answered by prakash jain last updated on 19/Feb/15
f(x+y)=f(x)+f(y)g(y) f(y+x)=f(y)+f(x)g(x) ....(1) g(x+y)=g(y)+f(x)g(x) ....(2) From (1) and (2) f(x)=g(x) f(0)=f(0)+[f(0)]^2 ⇒f(0)=0 f(x+y)=f(x)+[f(y)]^2 y=x, x=0 f(x)=[f(x)]^2 f(x)=1 or f(x)=0 f(x)=1 does not satisfy f(0)=0 hence f(x)=0
f(x+y)=f(x)+f(y)g(y)f(y+x)=f(y)+f(x)g(x)….(1)g(x+y)=g(y)+f(x)g(x)….(2)From(1)and(2)f(x)=g(x)f(0)=f(0)+[f(0)]2⇒f(0)=0f(x+y)=f(x)+[f(y)]2y=x,x=0f(x)=[f(x)]2f(x)=1orf(x)=0f(x)=1doesnotsatisfyf(0)=0hencef(x)=0

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