Question

If f and t are both even functions, is the product ft even? If f and t are both odd functions, is ft odd? What if f is even and t is odd? Justify your answers.

Answer 1

(a) If f and t are both even functions, product ft is even.

(b) If f and t are both odd functions, product ft is even.

(c) If f is even and t is odd, product ft is odd.

Explanation:

Even function: A function g(x) is called an even function if

`g(-x)=g(x)`

Odd function: A function g(x) is called an odd function if

`g(-x)=-g(x)`

(a)

Let f and t are both even functions, then

`f(-x)=f(x)`

`t(-x)=t(x)`

The product of both functions is

`ft(x)=f(x)t(x)`

`ft(-x)=f(-x)t(-x)`

`ft(-x)=f(x)t(x)`

`ft(-x)=ft(x)`

The function ft is even function.

(b)

Let f and t are both odd functions, then

`f(-x)=-f(x)`

`t(-x)=-t(x)`

The product of both functions is

`ft(x)=f(x)t(x)`

`ft(-x)=f(-x)t(-x)`

`ft(-x)=[-f(x)][-t(x)]`

`ft(-x)=ft(x)`

The function ft is even function.

(c)

Let f is even and t odd function, then

`f(-x)=f(x)`

`t(-x)=-t(x)`

The product of both functions is

`ft(x)=f(x)t(x)`

`ft(-x)=f(-x)t(-x)`

`ft(-x)=[f(x)][-t(x)]`

`ft(-x)=-ft(x)`

The function ft is odd function.