### Home > CALC > Chapter 2 > Lesson 2.3.3 > Problem2-125

2-125.

Determine if the following functions are even, odd, or neither. Explain how you determined your choice.

1. $y=\sin^2x$

$y=\sin^2x=\left(\sin x\right)\left(\sin x\right)$

$y=\sin x$ is an odd function. Definition of odd functions: $f\left(−x\right) = −f\left(x\right)$ Therefore, $\sin\left(−x\right)=−\sin\left(x\right)$.

Explore what $y=\sin^2\left(−x\right)$ looks like:
$y=\sin^2\left(−x\right)$
$=\left(\sin\left(−x\right)\right)\left(\sin\left(−x\right)\right)$
$=\left(−\sin x\right)\left(−\sin x\right)$
$=\sin^2x$

Hence $\sin^2\left(−x\right)=\sin^2\left(x\right)$.
This is the definition of an even function: $f\left(−x\right) = f\left(x\right)$.

1. $y = \frac { x ^ { 2 } + 1 } { x ^ { 3 } - 2 x }$

Test for even: $f\left(a\right) = f\left(−a\right)$

$f(a)=\frac{a^{2}+1}{a^{3}-2a}$

$f(-a)=\frac{(-a)^{2}+1}{(-a)^{3}-2(-a)}=\frac{a^{2}+1}{-a^{3}+2a}\ ,$

$f\left(a\right) ≠ f\left(−a\right)$; not even

Test for odd: $f\left(−a\right) = −f\left(a\right)...$