# How do you know if #v(x) = 2 sin x cos x# is an even or odd function?

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To determine if the function ( v(x) = 2 \sin(x) \cos(x) ) is even or odd, we need to check whether it satisfies the properties of even or odd functions.

- Even functions satisfy the property ( f(-x) = f(x) ) for all ( x ) in their domain.
- Odd functions satisfy the property ( f(-x) = -f(x) ) for all ( x ) in their domain.

For the function ( v(x) = 2 \sin(x) \cos(x) ), let's check these properties:

- Substitute ( -x ) into the function:

[ v(-x) = 2 \sin(-x) \cos(-x) ]

Since sine and cosine are both even functions, ( \sin(-x) = -\sin(x) ) and ( \cos(-x) = \cos(x) ).

So,

[ v(-x) = 2 (-\sin(x)) \cos(x) = -2 \sin(x) \cos(x) ]

This does not equal ( v(x) ), so the function is not even.

- Now, let's check if it's odd:

[ v(-x) = -2 \sin(x) \cos(x) ]

This is equal to ( -v(x) ), so the function satisfies the property of odd functions.

Therefore, ( v(x) = 2 \sin(x) \cos(x) ) is an odd function.

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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.

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.

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.

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