Question

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)]/2
b) h(x)= [f(x) – f(-x)]/2

Answer 1

Answer:

a.Even

b.Odd

Step-by-step explanation:

We are given that function f has domain all real numbers .

We have to find that given function is even or odd

a.$g(x)=\frac{f(x)+f(-x)}{2}$

We know that if function is even then f(x)=f(-x) and if function is odd then $f(x)\neq f(-x)$

Replace x by -x then we get

$g(-x)=\frac{f(-x)+f(x)}{2}=g(x)$

Therefore, function is even .

b.$h(x)=\frac{f(x)-f(-x)}{2}$

Replace x by -x

Then we get $h(x)=\frac{f(-x)-f(x)}{2}=-\frac{f(x)-f(-x)}{2}$

$h(x)\neq h(-x)$

Hence, function is odd.