Respuesta :
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.
Step-by-step explanation:
Even function: A function g(x) is called an even function if
[tex]g(-x)=g(x)[/tex]
Odd function: A function g(x) is called an odd function if
[tex]g(-x)=-g(x)[/tex]
(a)
Let f and t are both even functions, then
[tex]f(-x)=f(x)[/tex]
[tex]t(-x)=t(x)[/tex]
The product of both functions is
[tex]ft(x)=f(x)t(x)[/tex]
[tex]ft(-x)=f(-x)t(-x)[/tex]
[tex]ft(-x)=f(x)t(x)[/tex]
[tex]ft(-x)=ft(x)[/tex]
The function ft is even function.
(b)
Let f and t are both odd functions, then
[tex]f(-x)=-f(x)[/tex]
[tex]t(-x)=-t(x)[/tex]
The product of both functions is
[tex]ft(x)=f(x)t(x)[/tex]
[tex]ft(-x)=f(-x)t(-x)[/tex]
[tex]ft(-x)=[-f(x)][-t(x)][/tex]
[tex]ft(-x)=ft(x)[/tex]
The function ft is even function.
(c)
Let f is even and t odd function, then
[tex]f(-x)=f(x)[/tex]
[tex]t(-x)=-t(x)[/tex]
The product of both functions is
[tex]ft(x)=f(x)t(x)[/tex]
[tex]ft(-x)=f(-x)t(-x)[/tex]
[tex]ft(-x)=[f(x)][-t(x)][/tex]
[tex]ft(-x)=-ft(x)[/tex]
The function ft is odd function.
