Odd Function Times Even Function is Odd
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Theorem
Let $X \subset \R$ be a symmetric set of real numbers:
- $\forall x \in X: -x \in X$
Let $f: X \to \R$ be an odd function.
Let $g: X \to \R$ be an even function.
Let $f \cdot g$ denote the pointwise product of $f$ and $g$.
Then $\paren {f \cdot g}: X \to \R$ is an odd 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 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {-\map f x} \cdot \map g x\) | Definition of Odd Function and Definition of Even Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\map f x \cdot \map g x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -\map {\paren {f \cdot g} } x\) | Definition of Pointwise Multiplication of Real-Valued Functions |
Thus, by definition, $\paren {f \cdot g}$ is an odd function.
$\blacksquare$