one of my friends is trying to convinve me that this is true in math?

Question

he's telling me that if functions f and g are both odd, then so is f(g) f+g and f X g.
Can anyone prove or disprove this so that I can decide wether or not to put money on this bet. Thanks

Answer 1

If f and g are odd, then f∘g and f+g are also odd, but fg is not odd -- in fact, it is even.

Proof:

Let f and g be odd -- i.e. f(-x) = -f(x) and g(-x) = -g(x). Then (f∘g)(-x) = f(g(-x)) = f(-g(x)) = -f(g(x)) = -(f∘g)(x), so f∘g is odd. (f+g)(-x) = f(-x) + g(-x) = -f(x) + (-g(x)) = -(f(x) + g(x)) = -(f+g)(x), so f+g is odd. However, (fg)(-x) = f(-x)g(-x) = (-f(x))(-g(x)) = f(x)g(x) = (fg)(x), so fg is even.

Answer 2

for example if f= x and g = x, they both odd right? but f*g = x^2 it's even. so it is wrong. but for f(g) or f + g, i think it is right to say that.