Question

let f be a function such that f( x + f(y))= f(x) + y.
then find f(0). where x and y belongs to R

Answer 1

We can infer that f(0) = 0, for that if we set y=0, then the expression became f(x + f(0) ) = f(x) + 0, and then f(0) must be 0. ( However, a special case can occur (Too bad) : if f(x) is periodic of period f(0), then it is possible that f(x + f(0) = f(x) even if f(0) != 0. Depending on the context, you should or should'nt take this case into consideration.)

Answer 2

i got f(0)=0
at x=0 and y=0
f(0+f(0))=f(0)+0
f(f(0))=f(0)
so f(0)=0..
do i missed something in solution???

Answer 3

Yes. For example, lets say that f(0) some value K. The value at point A is K, and so is it at B. A and B are separated by the distance f(0), and the equation f(x + f(0) ) = f(x) + 0 holds, and however, f(0) != 0. What is your age /level of education ? (Because that it a rather complicated case, and perhaps you are not required to know it.)