Answer
The product of two even functions is an even function.
The product of two odd functions is an even function.
The product of an even function and an odd function is an odd function.
Work Step by Step
Suppose both $f$ and $g$ are even functions.
We can find an expression for $f(-x)~g(-x)$:
$f(-x)~g(-x) = f(x)~g(x)$
So the product of two even functions is an even function.
Suppose both $f$ and $g$ are odd functions.
We can find an expression for $f(-x)~g(-x)$:
$f(-x)~g(-x) = [-f(x)]~[-g(x)] = f(x)~g(x)$
So the product of two odd functions is an even function.
Suppose $f$ is an even function and $g$ is an odd function.
We can find an expression for $f(-x)~g(-x)$:
$f(-x)~g(-x) = [f(x)]~[-g(x)] = -f(x)~g(x)$
So the product of an even function and an odd function is an odd function.