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

Aug 5, 2015

That function is even.

#### Explanation:

We will need to recall that $\sin x$ is odd.
That is: $\sin \left(- x\right) = - \sin x$

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

So

For any $x$ in the domain of $f$ (which is $\left(- \infty , \infty\right)$),
we get

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

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

$= x \sin x$

$= f \left(x\right)$

By the definition of even function, $f$ is even.