# Is the function f(x) = x^3 + x sin^2 x even, odd or neither?

Aug 16, 2015

$f \left(x\right)$ is odd

#### Explanation:

A function is even if if exhibits the property $f \left(- x\right) = f \left(x\right)$
A function is odd if it exhibits the property $f \left(- x\right) = - f \left(x\right)$

Let check for $f \left(x\right)$:
$f \left(- x\right)$
$= {\left(- x\right)}^{3} + \left(- x\right) {\sin}^{2} \left(- x\right)$
$= - {x}^{3} - x {\sin}^{2} x$
$= - f \left(x\right)$

Thus, $f \left(x\right)$ is odd. You can confirm this by graphing. Since ${x}^{3}$ is odd and $x {\sin}^{2} x$ is odd, therefore $f \left(x\right) = {x}^{3} + x {\sin}^{2} x$ is odd.

graph{x*(sin(x))^2 [-10.32, 10.295, -5.155, 5.155]}