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Math Help - Crazy Calculus Project

  1. #1
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    Crazy Calculus Project

    This is my crazy calculus professor coming up with this stuff.

    He wants us to show that if f is any function, then it can be written as the sum of even and odd functions. That is, if f[even] is an even function and
    f[odd] is an odd function, then:
    f(x) = f[even](x) + f[odd](x)
    He gives an example.
    Example:
    For f(x) = 1/(1+x), we can take f[even](x)=1/(1-x^2) and f[odd](x)=-x/(1-x^2).
    You can check for yourself that f[even](x) is in fact even and that f[odd](x) is indeed odd. And also you can see that:

    f[even]+f[odd]=[1/(1-x^2)] +[-x/(1-x^2)]=(1-x)/(1-x^2)=1/(1+x)=f(x)

    Even with this example, I am still confused as to how to show....(see paragraph one)
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  2. #2
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    Consider these functions.
    g(x) = \frac{{f(x) + f( - x)}}{2}\quad \& \quad h(x) = \frac{{f(x) - f( - x)}}{2}<br />
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  3. #3
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    How about this, PLATO.

    What if i said:
    Say f(x)=(x^3+2x^2-x-1)/(x^2-4)
    let g(x)=f[even](x)=(2x^2-1)/(x^2-4)
    and let h(x)=f[odd](x)=(x^3+x)/(x^2-4)
    Then g(x)+h(x)=f(x) am I right?
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  4. #4
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    Do you understand that your professor is asking you to show:
    Any function f can be written as the sum of an even function and an odd function?
    g(x) = \frac{{f(x) + f( - x)}}{2}\quad  \Rightarrow \quad g(x) = g( - x)\mbox{  even}
    h(x) = \frac{{f(x) - f( - x)}}{2}\quad  \Rightarrow \quad h(x) =  - h( - x)\mbox{  odd}
    f(x) = g(x) + h(x)

    Now that is true for Any function f.
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