Chapter 0.2, Problem 72E

$\left(\text{a}\right)$

To determine

The product of two even functions.

Answer to Problem 72E

The product of two even functions is always even.

Explanation of Solution

Given information:

Must the product of two even functions always be even. Give reason for answer.

Consider two even functions $f\left(x\right)$ and $g\left(x\right)$ :

Let $h\left(x\right)$ be their product. Now, check for their product to be even or odd as follows:

  $\begin{array}{l}h\left(x\right)=f\left(x\right)\times g\left(x\right)\\ h\left(-x\right)=f\left(-x\right)\times g\left(-x\right)\\ \Rightarrow =f\left(x\right)\times g\left(x\right)\\ \Rightarrow =h\left(x\right)\end{array}$

Therefore, the product of two even functions is always even.

  $\left(\text{b}\right)$

To determine

The product of two odd functions.

Answer to Problem 72E

The product of two odd functions is always odd.

Explanation of Solution

Given information:

Can anything be said about the product of two odd functions. Give reason for answer.

Consider two odd functions $f\left(x\right)$ and $g\left(x\right)$ :

Let $h\left(x\right)$ be their product. Now, check for their product to be even or odd as follows:

  $\begin{array}{l}h\left(x\right)=f\left(x\right)\times g\left(x\right)\\ h\left(-x\right)=f\left(-x\right)\times g\left(-x\right)\\ \Rightarrow =-f\left(x\right)\times -g\left(x\right)\\ \Rightarrow =f\left(x\right)\times g\left(x\right)\\ \Rightarrow =h\left(x\right)\end{array}$

Therefore, the product of two odd functions is always odd.