Q:

Suppose that the function f has domain all real numbers. Determine whether each function can be classified as even or odd. Explain.a) g(x)= [f(x) + f(-x)]/2b) h(x)= [f(x) – f(-x)]/2

Accepted Solution

A:
Answer:a.Even b.OddStep-by-step explanation:We are given that function f has domain all real numbers .We have to find that given function is even or odda.[tex]g(x)=\frac{f(x)+f(-x)}{2}[/tex]We know that if function is even then f(x)=f(-x) and if function is odd then [tex]f(x)\neq f(-x)[/tex]Replace x by -x then we get [tex]g(-x)=\frac{f(-x)+f(x)}{2}=g(x)[/tex]Therefore, function is even .b.[tex]h(x)=\frac{f(x)-f(-x)}{2}[/tex]Replace x by -xThen we get [tex]h(x)=\frac{f(-x)-f(x)}{2}=-\frac{f(x)-f(-x)}{2}[/tex][tex]h(x)\neq h(-x)[/tex]Hence, function is odd.