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Determine all functions `f: Rvecs u c ht h a tf(x-f(y))=f(f(y))+xf(y)+f(x)-1AAx , ygeq in Rdot` |
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Answer» Correct Answer - `f(x)=1-(x^(2))/(2)` Given `f(x-f(y))=f(f(y))+xf(y)+f(x)-1 " (1) " ` Putting `x=f(y)=0,` we get `f(0)=f(0)+0+f(0)-1,` i.e., `f(0)=1 " (2)" ` Again, putting `x=f(y)= lambda` in (1), we get `f(0)=f(lambda)+lambda^(2)+f(lambda)-1` or `1=2f(lambda)+lambda^(2)-1 " [From (2)]" ` `f(lambda)=(2-lambda^(2))/(2)=1-(lambda^(2))/(2)` Hence, `f(x)=1-(x^(2))/(2).` |
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