Let x = 0. Then the equation becomes f(0) + y = f(y) for every real number y.

f(0) is just a constant, so write c = f(0). Then, we get f(y) = y+c for every real number y.

Thus, the only possible functions that work are functions of the form f(x) = x+c. We can easily check that this function works. So the answer is "f(x) = x+c for some constant c."