# Question #93699

Feb 3, 2017

$f \left(x\right) = x \sin \left(x\right)$ is even.

#### Explanation:

We can tell if a function $f \left(x\right)$ is even or odd by examining what happens to $f \left(- x\right)$. If $f \left(- x\right) = f \left(x\right)$, then it is even. If $f \left(- x\right) = - f \left(x\right)$, then it is odd. It is possible for a function to be both even and odd (in the case where $f \left(x\right) = 0$) or neither even nor odd.

For the function in question, noting that $\sin \left(- x\right) = - \sin \left(x\right)$, we have

$f \left(- x\right) = \left(- x\right) \sin \left(- x\right)$

$= - x \left(- \sin \left(x\right)\right)$

$= x \sin \left(x\right)$

$= f \left(x\right)$.

Thus the function is even.