Even and Odd Functions Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

Prove that the product of two odd functions is an even function.

1
Let $f$ and $g$ be odd: $f(-x) = -f(x)$ and $g(-x) = -g(x)$ . Let $h(x) = f(x) \cdot g(x)$ .
2
$h(-x) = f(-x) \cdot g(-x) = (-f(x))(-g(x)) = f(x) \cdot g(x) = h(x)$ . Since $h(-x) = h(x)$ , $h$ is even.

Answer

\text{Proven: the product of two odd functions is even.}

The algebra of even and odd functions follows rules similar to the signs of integers: even

\times

even

=

even, odd

\times

odd

=

even, even

\times

odd

=

odd. This mirrors the exponent rule: multiplying two odd-powered terms gives an even-powered term.

About Even and Odd Functions

An even function satisfies $f(-x) = f(x)$ (symmetric about $y$ -axis); an odd function satisfies $f(-x) = -f(x)$ (rotational symmetry about origin).

Determine whether [formula] is even, odd, or neither.

Determine whether [formula] is even, odd, or neither.

Is [formula] even, odd, or neither?