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Homework Help: Analysis Proof: Sum of odd and even functions.

  1. Apr 8, 2010 #1
    1. The problem statement, all variables and given/known data
    Show f(x) can be expressed as the sum of E(x) and an odd function O(x).

    f(x) is defined for all x (assume domain D symmetric about 0)

    [tex]f(x) = \frac{f(x)+f(-x}{2}[/tex]
    How does it look if [tex]f(x) = e^x[/tex]?

    2. Relevant equations

    3. The attempt at a solution

    So I got [tex]\frac{f(x)+f(-x)+f(x)-f(-x)}{2}[/tex]

    as my solution for that.

    From that I need to answer: How does it look if [tex]f(x) = e^x[/tex] ? I'm not sure what to do to show that.
  2. jcsd
  3. Apr 8, 2010 #2

    Char. Limit

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    Gold Member

    First: Is that a sum of odd and even functions?

    Second: Well, try sticking e^x in for f(x) and see what functions you get.
  4. Apr 8, 2010 #3
    I stuck in e^x for f(x) and got e^x. Is this supposed to represent something?
  5. Apr 8, 2010 #4

    Char. Limit

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    Gold Member

    You should get...

  6. Apr 8, 2010 #5


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    Well, since the two sides are equal I would hope you get ex back when you cancel everything. But notice without canceling what you get

    [tex]e^x = \frac{e^x+e^{-x}}{2} + \frac{e^x-e^{-x}}{2} = cosh(x)+sinh(x)[/tex] which is a way of writing [tex]e^x[/tex] as a sum of an odd and an even function
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