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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.

User PKay
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Answer:

(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.

User Androo
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