# To prove : that if f and g are both even and odd function, then the product, of if one is odd and one is even ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071 ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 2.5, Problem 92E
To determine

## To prove : that if f and g are both even and odd function, then the product, of if one is odd and one is even

Expert Solution

### Explanation of Solution

Given information : The question is

f·g Necessarily even and if both are odd, is their product necessarily odd

Proof : Let two even functions f&g . Since these function are even

f(x)=f(x)&g(x)=g(x)

The product of these two functions is defined as

(f·g)(x)=f(x)g(x)(f·g)(x)=f(x)g(x)(f·g)(x)=f(x)g(x)(f·g)(x)=(f·g)(x)

The product of two even functions is also an even function

Let two odd functions f&g . Since these function are odd

f(x)=f(x)&g(x)=g(x)

The product of these two functions Is defined as

(f·g)(x)=f(x)g(x)(f·g)(x)=f(x)g(x)(f·g)(x)=[f(x)][g(x)](f·g)(x)=f(x)g(x)(f·g)(x)=(f·g)(x)

The product of two odd functions is an even function

Let an even function f and an odd function g

f(x)=f(x)&g(x)=g(x)

The product of these two functions is defined as

(f·g)(x)=f(x)g(x)(f·g)(x)=f(x)g(x)(f·g)(x)=f(x)[g(x)](f·g)(x)=f(x)g(x)(f·g)(x)=(f·g)(x)

The product of an even and an odd function is an odd function

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