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Math Help - Odd extension, even extension?

  1. #1
    Super Member fardeen_gen's Avatar
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    Odd extension, even extension?

    Let the function f(x) = x^2  + x + \sin x - \cos x + \ln(1 + |x|) be defined over the interval [0,1]. Find the odd and even extension of f(x) in the interval [-1, 1]

    Answer:
    Spoiler:
    \mbox{Odd extension:}\ -x^2 + x + \sin x + \cos x - \ln(1 + |x|) \ <br />
\mbox{Even extension:}\ x^2 - x - \sin x - \cos x + \ln(1 + |x|)


    How to do it?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by fardeen_gen View Post
    Let the function f(x) = x^2 + x + \sin x - \cos x + \ln(1 + |x|) be defined over the interval [0,1]. Find the odd and even extension of f(x) in the interval [-1, 1]

    Answer:
    Spoiler:
    \mbox{Odd extension:}\ -x^2 + x + \sin x + \cos x - \ln(1 + |x|) \
    Spoiler:
    \mbox{Even extension:}\ x^2 - x - \sin x - \cos x + \ln(1 + |x|)" alt="
    \mbox{Even extension:}\ x^2 - x - \sin x - \cos x + \ln(1 + |x|)" />


    How to do it?
    The odd extension f(x) on [-1,0) satisfies:

    g(x)=-f(-x)

    and the even extension:

    h(x)=f(-x).

    CB
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