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Math Help - Proving any function can be written in the form f = O + E ?

  1. #1
    Member mfetch22's Avatar
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    Proving any function can be written in the form f = O + E ?

    If E is any function such that:

    E(x) = E(-x)

    And O is any function such that:

    -O(x) = O(-x)

    Then prove that any given function f can be expressed such that:

    f(x) = E(x) + O(x)

    Now, I will show what work I've done and see if anybody can tell me if I'm on the right track. Lets assume firstly that f is even. Let h be any choosen odd function and g be any choosen even function; then we can set:

    f = g

    Then we know that f can be written as:

    f(x) = g(x) = f(-x) = g(-x)

    Now, if we let:

    h(x) = 0

    then:

    h(x) = h(-x) = -h(x)

    So h is either even or odd, so if f is even we can write it as:

    f(x) = g(x) + h(x) = g(x) + 0 = g(x) = f(x)

    Assuming f is odd then we can set:

    f(x) = h(x)

    And simmiliarly to the above:

    g(x) = 0

    So we know g can be either even or odd. So we let g stand in for our even function, and we can write f as:

    f(x) = h(x) + g(x) = h(x) + 0 = h(x) = f(x)

    But, how do I now prove the cases where we assume f(x) = 0 and when f is not even or odd? Any guidance would be appreciated. Thanks
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  2. #2
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    Separation into cases is not needed --
    You can write \displaystyle f(x) = \frac{f(x)+f(-x) -f(-x)+f(x)}{2} = \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2}
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  3. #3
    Member mfetch22's Avatar
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    Quote Originally Posted by Defunkt View Post
    Separation into cases is not needed --
    You can write \displaystyle f(x) = \frac{f(x)+f(-x) -f(-x)+f(x)}{2} = \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2}
    Wait, did u assume f to be even or odd in the above work? Or did you simply manipulate it into an equal form which represents the ability to equate the "even and odd" forms of f?
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  4. #4
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    Quote Originally Posted by mfetch22 View Post
    Wait, did u assume f to be even or odd in the above work? no Or did you simply manipulate it into an equal form which represents the ability to equate the "even and odd" forms of f? correct
    .
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