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.
Accepted Solution
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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.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.