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Math Help - Determine whether function is even or odd

  1. #1
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    Determine whether function is even or odd

    Hi!

    Determine whether the function is even, odd or neither.

    1. f(x) = x^3 \sin x

    The function is even since:

    f(-x) = -x^3\sin(-x) = x^3sinx

    The result of cubing a negative is negative and taking the sine of a negative gives a negative, and multiplying two negatives gives a positive.
    Correct any flaws in this thinking process please!

    2. f(x) = \cos(\sinx)

    This function is even since:

    f(-x) = \cos(\sin(-x) = \cos(-\sin(x)) = -\cos(\sin(x) = \cos(\sinx)
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  2. #2
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    Re: Determine whether function is even or odd

    Quote Originally Posted by Unreal View Post
    Hi!

    Determine whether the function is even, odd or neither.

    1. f(x) = x^3 \sin x

    The function is even since:

    f(-x) = -x^3\sin(-x) = x^3sinx

    The result of cubing a negative is negative and taking the sine of a negative gives a negative, and multiplying two negatives gives a positive.
    Correct any flaws in this thinking process please!

    2. f(x) = \cos(\sinx)

    This function is even since:

    f(-x) = \cos(\sin(-x) = \cos(-\sin(x)) = -\cos(\sin(x) = \cos(\sinx)
    The first is correct. For the second, recall that cos(-x) = cos(x). Cosine is an even function, not odd.

    -Dan
    Thanks from Unreal
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  3. #3
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    Re: Determine whether function is even or odd

    f(-x) = \cos(\sin(-x) = \cos(-\sin(x)) = -\cos(\sin(x) = \cos(sinx)
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    Forum Admin topsquark's Avatar
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    Re: Determine whether function is even or odd

    Quote Originally Posted by Unreal View Post
    f(-x) = \cos(\sin(-x) = \cos(-\sin(x)) = -\cos(\sin(x) = \cos(sinx)
    f(-x) = cos( sin( -x )) = cos( -sin(x) ) = cos( sin(x) ) = f(x)

    Thus f(x) is even.

    -Dan
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  5. #5
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    Re: Determine whether function is even or odd

    Quote Originally Posted by topsquark View Post
    The first is correct. For the second, recall that cos(-x) = cos(x). Cosine is an even function, not odd.

    -Dan
    The first is NOT correct (or at least is incomplete), as the OP should have written \displaystyle \begin{align*}  f(-x) = (-x)^3\sin{(-x)} \end{align*}.
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