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.