Chapter 1.5, Problem 91E

To determine

To prove: product of an odd function and an even function is even or odd based on the example used to hypothesize whether the product of an odd function and an even function is even or odd.

Explanation of Solution

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

Proof:

First take an example of an odd and an even function

Let $f(x)=x$ and $g(x)=x^2$

Here,

  $\begin{array}{l}f(-x)=-x=-f(x)\\ $ g(-x)={(-x)}^2=x^2=g(x)$\end{array}$

So, $f$ is an odd function and $g$ is an even function.

Now find product of functions $f$ and $g$

  $(fg)(x)=f(x)g(x)=(x)(x^2)=x^3$

To check whether product of functions $f$ and $g$ is even or odd, find $\left(fg\right)(-x)$

  $(fg)(-x)={(-x)}^3=-x^3=-(fg)\left(x\right)$

So, $fg$ is an odd function.

Now prove the hypothesis.

Let $f$ be an odd function and $g$ be an even function.

So,

  $f(-x)=-f(x),\text{}g(-x)=g(x)$

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

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

So, $fg$ is an odd function.