Symmetry of product of symmetric functions

Question

Let's say we have three functions, $f(x), g(x) \text{ and }h(x)=f(x)\cdot g(x)$, all of which are defined for $x \in [-1,1]$. If $f(x) \text{ and }g(x)$ are symmetric around the y-axis, will $h(x)$ also be symmetric around the y-axis?

Answer 1

$f$, $g$ being symmetric about the $y$-axis means $f(x)=f(-x)$ and $g(x)=g(-x)$. Then also

$$h(x)=f(x)g(x)=f(-x)g(-x)=h(-x)$$

so $h$ is also symmetric about the $y$-axis.

Answer 2

Easy to check :

$h(-x)=f(-x)g(-x)=f(x)g(x)=h(x)$

So yes.