Is the function $f (x) = x sin x$ $f(x) =x sin x$ 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 $sin x$ $sinx$ is odd.
That is: $sin (- x) = - sin x$ $sin(-x) = -sinx$

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

So

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

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

$= (- x) (- sin x)$ $= (-x)(-sinx)$

$= x sin x$ $= xsinx$

$= f (x)$ $= f(x)$

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

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