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Thread: 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 $\displaystyle E$ is any function such that:

    $\displaystyle E(x) = E(-x) $

    And $\displaystyle O$ is any function such that:

    $\displaystyle -O(x) = O(-x)$

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

    $\displaystyle f(x) = E(x) + O(x)$

    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 $\displaystyle f$ is even. Let $\displaystyle h$ be any choosen odd function and $\displaystyle g$ be any choosen even function; then we can set:

    $\displaystyle f = g$

    Then we know that f can be written as:

    $\displaystyle f(x) = g(x) = f(-x) = g(-x)$

    Now, if we let:

    $\displaystyle h(x) = 0$

    then:

    $\displaystyle h(x) = h(-x) = -h(x)$

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

    $\displaystyle f(x) = g(x) + h(x) = g(x) + 0 = g(x) = f(x)$

    Assuming $\displaystyle f$ is odd then we can set:

    $\displaystyle f(x) = h(x)$

    And simmiliarly to the above:

    $\displaystyle g(x) = 0$

    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:

    $\displaystyle f(x) = h(x) + g(x) = h(x) + 0 = h(x) = f(x)$

    But, how do I now prove the cases where we assume $\displaystyle f(x) = 0$ and when $\displaystyle f$ 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 $\displaystyle \displaystyle f(x) = \frac{f(x)+f(-x) -f(-x)+f(x)}{2} = \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2}$
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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 $\displaystyle \displaystyle f(x) = \frac{f(x)+f(-x) -f(-x)+f(x)}{2} = \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2}$
    Wait, did u assume $\displaystyle f$ 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 $\displaystyle f$?
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  4. #4
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    Quote Originally Posted by mfetch22 View Post
    Wait, did u assume $\displaystyle f$ 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 $\displaystyle f$? correct
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