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

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Answer 1 · 2015-08-05T01:09:42

That function is even.

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

$f(x) =x sin x$

So

For any $x$ in the domain of $f$ (which is $(-oo,oo)$ ),
we get

$f(-x) =(-x) sin (-x)$

$= (-x)(-sinx)$

$= xsinx$

$= f(x)$

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

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