Answer
1) If both $f$ and $g$ are even $fg$ is even;
2) If both $f$ and $g$ are odd $fg$ is even;
3) If $f$ is even and $g$ is odd $fg$ is odd.
Work Step by Step
The new function of some argument $x$ is the product:
$$h(x)=f(x)g(x).$$
To say when it is odd and when it is even we have to check what happens when we substitute $x$ for $-x$. Check all the combinations:
1) $f$ is even and $g$ is even. Then $f(-x)=f(x)$ and $g(-x)=g(x)$ so we have:
$$h(-x)=f(-x)g(-x)=f(x)g(x)=h(x)$$
so we have $h(-x)=h(x)$ and we see that $h$ is even.
2) $f$ is odd and $g$ is odd. Then $f(-x)=-f(x)$ and $g(-x)=-g(x)$ so we have:
$$h(-x)=f(-x)g(-x)=(-f(x))\cdot(-g(x))=f(x)g(x)=h(x)$$
so we have $h(-x)=-h(x)$ and we see that $h$ is even.
3) $f$ is even and $g$ is odd. Then $f(-x)=f(x)$ and $g(-x)=-g(x)$ so we have:
$$h(-x)=f(-x)g(-x)=f(x)\cdot(-g(x))=-h(x)$$
so we have $h(-x)=-h(x)$ and we see that $h$ is odd.
