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Continuous function

  1. Mar 22, 2004 #1
    is the following statement ture?

    A continuous function f(x) can be rewritten as f(x)=E(x)+ O(x)
    where:
    1) E(x)is even function and O(x) is odd function.
    2) Both E(x),O(x) are continuous function too
    3) f,E,O are defined in (-oo,oo)

    my classmate say it is ture but can not prove it
    i think it can't be right since i can't figure out a way to rewrite a power function to E(x)+ O(x). eg e^x

    any good sugguestion?
    thx
     
  2. jcsd
  3. Mar 22, 2004 #2

    matt grime

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    Can you show f(x)+f(-x) even?

    Can you generate an odd function like it?

    Can you now recover f?
     
  4. Mar 22, 2004 #3
    yeah,this one is easy

    hmmmm.....more hints plz
     
  5. Mar 22, 2004 #4

    matt grime

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    Here's a much better way of getting you to think of the answer for yourself:

    you wanted to do this for e^x?

    Do you know what hyperbolic trig functions are?

    cosh(x) is even, sinh(x) is odd

    cosh(x)+sinh(x) = e^x

    if you need to, look up these at, say, wolfram.

    If you need more just say, but it's always best to give you the means, especially if it's in terms of stuff you know, and it seems reasonable if you're doing continuity, that you know what cosh and sinh are.
     
  6. Mar 22, 2004 #5
    well.... i am not looking at any specific answer.
    i am trying to find out why it is true.

    thx for telling me that, i just check it out and it is true


    i am still following ur hint(i have played with it for 2hrs )


    if we let g(x)=f(-x)+f(x)
    then g(-x)=f(x)+f(-x)=g(x)....(1)
    therefore g is even

    rearrange (1) gives
    -f(x)=f(-x)-g(x).....(2)
    and
    f(-x)=g(x)-f(x)......(3)

    but it doesn't work
    what should i do next??
     
  7. Mar 22, 2004 #6

    matt grime

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    So for e^x we have

    cosh(x)=(e^(x)+e^(-x))/2

    and

    sinh(x) = (e^(x)-e^(-x))/2


    and remember f(x) = e^x here, and f(-x)=e^(-x)

    can you see how that generalizes?
     
  8. Mar 22, 2004 #7
    :frown: omg i just found out how stupid i am
    i wanna shoot myself....errr

    but really thx matt
    u r the man
     
  9. Mar 22, 2004 #8

    matt grime

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    Hope you think it's better to figure these things out some times than just be told them; I'm trying to stick to Polya's views on teaching.. reminds me to start a thread on that some time
     
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