Question

DISCUSS: Even and Odd Power Functions What must betrue about the integern if the function

$f\left(x\right)=x^n$

is an even function? If it is an odd function? Why do you think the names "even" and "odd" were chosen for these function properties?

Short answer

If$f\left(x\right)$is an even function then the integer n must be even.

If$f\left(x\right)$is an odd function then the integer n must be odd.

Ifthe function is even then$f\left(-x\right)=f\left(x\right)$that is ifn is even then $f\left(-x\right)=f\left(x\right)$.

If the function is odd then $f\left(x\right)=-f\left(x\right)$that is ifn is even then $f\left(-x\right)=-f\left(x\right)$.

Step-by-step solution

Step 1. Answer the first question.

If $f\left(x\right)$is an even function then the integer n must be even.

Step 2. Answer the second question.

If $f\left(x\right)$is an odd function then the integer n must be odd.

Step 3. Answer the third question.

Ifthe function is even then$f\left(-x\right)=f\left(x\right)$that is ifn is even then $f\left(-x\right)=f\left(x\right)$.

If the function is odd then $f\left(x\right)=-f\left(x\right)$that is ifn is even then $f\left(-x\right)=-f\left(x\right)$.