If F(x) + i*G(x) = J(x) + i*K(x), is it possible to prove that F(x) = J(x) and that i*G(x) = i*K(x)

Assuming that F, G, J, and K are real-valued, this is true.

Given F(x) + i*G(x) = J(x) + i*K(x):

Take conjugates of both sides:
F(x) - i*G(x) = J(x) - i*K(x).

Adding these equations yields
2 F(x) = 2 J(x) ==> F(x) = J(x).

Hence, F(x) + i * G(x) = F(x) + i * K(x) ==> G(x) = K(x).

I hope this helps!