Answer
(a) neither
(b) odd
(c) even
(d) neither
(e) even
(f) neither
Work Step by Step
Recall:
$f(x)$ is even if $f(-x)=f(x)$
$f(x)$ is odd if $f(-x)=-f(x)$
(a) $f(-x)=2(-x)^5-3(-x)^2+2$
$f(x)=2(-x^5)-3x^2+2$
$f(x)=-2x^5-3x^2+2$
$f(x)\neq \pm f(x)$
$\therefore f(x)$ is neither even nor odd.
(b) $f(-x)=(-x)^3-(-x)^7$
$f(x)=-x^3-(-x^7)$
$f(x)=-(x^3-x^7)$
$f(x)=-f(x)$
$\therefore f(x)$ is odd.
(c) $f(-x)=e^{-(-x)^2}$
$f(x)=e^{-x^2}$
$f(x)=f(x)$
$\therefore f(x)$ is even.
(d) $f(-x)=1+\sin(-x)$
$f(x)=1-\sin(x)$
$f(x)\neq \pm f(x)$
$\therefore f(x)$ is neither even nor odd.
(e) $f(-x)=1-\cos (2(-x))$
$f(x)=1-\cos(-2x)$
$f(x)=1-\cos(2x)$
$f(x)=f(x)$
$\therefore f(x)$ is even.
(f) $f(-x)=(-x+1)^2$
$f(x)=(-(x-1))^2$
$f(x)=(x-1)^2$
$f(x)\neq \pm f(x)$
$\therefore f(x)$ is neither even nor odd.
