Chapter 1.2, Problem 72E

(a)

To determine

To check: Whether the product of two even functions always even or not.

Answer to Problem 72E

Yes, the product of two even functions is always an even function.

Explanation of Solution

Given information: Consider two even functions.

Assume that two functions $f\left(x\right)$ and $g\left(x\right)$ are even functions. By the definition of even function $f\left(-x\right)=f\left(x\right)$ and $g\left(-x\right)=g\left(x\right)$ .

Now, check the property of even function for the product of both the functions.

  $\begin{array}{c}\left(f\cdot g\right)\left(-x\right)=\left(f\left(-x\right)\right)\cdot \left(g\left(-x\right)\right)\\ =\left(f\left(x\right)\right)\cdot \left(g\left(x\right)\right)\\ =\left(f\cdot g\right)\left(x\right)\end{array}$

So, the product of both the even functions is even.

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

(b)

To determine

To check: The type of the function of product of two odd functions.

Answer to Problem 72E

The product of two odd functions is always even function.

Explanation of Solution

Given information: Consider any two odd functions.

Assume that two function $f\left(x\right)$ and $g\left(x\right)$ are two odd functions. By the definition of odd function $f\left(-x\right)=-f\left(x\right)$ and $g\left(-x\right)=-g\left(x\right)$ .

Now, check the property of even function for the product of both the functions.

  $\begin{array}{c}\left(f\cdot g\right)\left(-x\right)=\left(f\left(-x\right)\right)\cdot \left(g\left(-x\right)\right)\\ =\left(-f\left(x\right)\right)\cdot \left(-g\left(x\right)\right)\\ =\left(f\left(x\right)\right)\cdot \left(g\left(x\right)\right)\\ =\left(f\cdot g\right)\left(x\right)\end{array}$

So, the product of both the odd functions is an even function.

Therefore, the product of two odd functions is always an even function.