Chapter 1.5, Problem 92E

To determine

To prove: product of two odd functions is an even function, and the product of two even functions is an even function

Explanation of Solution

Formula used: If $f$ is an even function then $f(-x)=f(x)$ and if $f$ is an odd function then $f(-x)=-f(x)$

Proof:

First prove that product of two odd functions is an even function

Let $f,g$ be two odd functions

So,

  $\begin{array}{l}f(-x)=-f(x)\\ g(-x)=-g(x)\end{array}$

Now find $\left(fg\right)(-x)$

  $\left(fg\right)(-x)=f(-x)g(-x)=\left[-f(x)\right]\left[-g(x)\right]=f(x)g(x)=(fg)(x)$

As $\left(fg\right)(-x)=(fg)(x)$ , $fg$ is an even function

Now prove that product of two even functions is an even function

Let $f,g$ be two even functions

So,

  $\begin{array}{l}f(-x)=f(x)\\ g(-x)=g(x)\end{array}$

Now find $\left(fg\right)(-x)$

  $\left(fg\right)(-x)=f(-x)g(-x)=f(x)g(x)=(fg)(x)$

As $\left(fg\right)(-x)=(fg)(x)$ , $fg$ is an even function