Suppose f and g are even. Then f(g(-x)) = f(g(x) so f(g) is even. If f and g are both odd, then f(g(-x)) = f(-g(x))=-f(g(x)), so f(g) is odd. You try the other two.
If f and g are defined for all x and are odd or even (four possibilities altogether), what can be said about the composition ? Prove it.
I'm not sure what can be said about the composition or what the four possiblilities should be.
Help is appreciated.