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How do you know if #x*sin x# is an even or odd function?

Answer 1

Even

A function f is even if the graph of f is symmetric with respect to the y-axis. Algebraically, f is even if and only if f(-x) = f(x) for all x in the domain of f.

Here is graph and note the symmetry with the y-axis :

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Answer 2

Ariana Alvarez

To determine if ( x \sin(x) ) is an even or odd function, evaluate ( f(-x) ) and compare it with ( f(x) ), where ( f(x) = x \sin(x) ).

If ( f(-x) = f(x) ), the function is even.

If ( f(-x) = -f(x) ), the function is odd.

Evaluate ( f(-x) ):

[ f(-x) = (-x) \sin(-x) = -x \sin(x) ]

Compare ( f(-x) ) with ( f(x) ):

[ f(-x) = -f(x) ]

Therefore, ( x \sin(x) ) is an odd function.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.