the prophet sir ke liye sawal

You cannot assume that.

f(x) = g(x) = 0 for all x is also a solution.

Assuming that is not the case, setting x=y=0, we have

f(0) = 2f(0) g(0) and g(0) = g²(0) - f²(0)

Either f(0) = 0 or g(0) = 1/2.

But if g(0) = 1/2, then Eqn 2 gives f²(0) = -1/2 which is not possible.

So, f(0) = 0.

From the second equation, g(0)(1-g(0)) = 0, so g(0) = 0 or g(0) =1.

Now, if g(0) =0, we have setting y=0 in the equations, f(x)=0 for all x and so g(x) for all x which we have already noted as a solution.

So we have the only other possibility as f(0) =0 and g(0) = 1

@EmInEm
u cannot assume that!

thank you sir for the solution[1]