Even Function Times Even Function is Even

Theorem

Let $X \subset \R$ be a symmetric set of real numbers:

$\forall x \in X: -x \in X$

Let $f, g: X \to \R$ be two even functions.

Let $f \cdot g$ denote the pointwise product of $f$ and $g$.

Then $\paren {f \cdot g}: X \to \R$ is also an even function.

Proof

\(\ds \map {\paren {f \cdot g} } {-x}\)

\(=\)

\(\ds \map f {-x} \cdot \map g {-x}\)

Definition of Pointwise Multiplication of Real-Valued Functions

\(\ds \)

\(=\)

\(\ds \map f x \cdot \map g x\)

Definition of Even Function

\(\ds \)

\(=\)

\(\ds \map {\paren {f \cdot g} } x\)

Definition of Pointwise Multiplication of Real-Valued Functions

Thus, by definition, $\paren {f \cdot g}$ is an even function.

$\blacksquare$

Also see

• Odd Function Times Even Function is Odd
• Odd Function Times Odd Function is Even