Exercise 1.1.88

Question

If $f$ and $g$ are both even functions, is the product $fg$ even? If $f$ and $g$ are both odd functions, is $fg$ odd? What if $f$ is even and $g$ is odd? Justify your answers.

Khagan Gulusoy · Accepted Answer

Recall that $$\forall x\in X:\quad fg(x)=f(x)\cdot g(x),$$ where $X$ is the domain of the function $fg$
    \begin{itemize}
    \item (if $f$ and $g$ are both even functions, is $fg$ even) We have, for all $x\in X$:
    $$(fg)(-x)=f(-x)\cdot g(-x)=f(x)\cdot g(x)=fg(x).$$
    Thus, $fg$ is even.
    \item (if $f$ and $g$ are both odd functions, is $fg$ odd) We have, for all $x\in X$:
    $$fg(-x)=f(-x)\cdot g(-x)=-f(x)\cdot (-g(x))=f(x)\cdot g(x)=fg(x).$$
    Thus, $fg$ is even.
\end{itemize}

Exercise 1.1.88

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Exercise 1.1.88

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