Even & odd functions

Discussion in 'Homework Help' started by Asad1, Feb 11, 2009.

  1. Asad1

    Thread Starter Member

    Feb 11, 2009
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    why an odd function always start with zero? :confused::confused::confused:
     
  2. Ratch

    New Member

    Mar 20, 2007
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    Asad1,

    What do you mean? Be definition, the function starts in the negative region of the x-axis for both odd and even functions. Perhaps an example is in order to understand what you are asking.

    Ratch
     
  3. Alexei Smirnov

    Active Member

    Jan 7, 2009
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    Odd function always = 0 at zero: f(0) = 0
     
  4. Ratch

    New Member

    Mar 20, 2007
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    Alexei Smirnov,

    f(x) = 1/x is a odd function which is not defined at 0.

    Ratch
     
  5. Alexei Smirnov

    Active Member

    Jan 7, 2009
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    Ok, ok :)
    f(-x)=-f(x)...
    only continuous function=0...
     
  6. Asad1

    Thread Starter Member

    Feb 11, 2009
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    Actually i need a mathematical and analytical prove that an odd signal have zero at the origin. Like sine is an odd function and it starts with Zero, The question is that why an odd function always get start from 0.
     
    Last edited: Feb 11, 2009
  7. Alexei Smirnov

    Active Member

    Jan 7, 2009
    43
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    f(-x)=-f(x); (definition of odd function)
    f(-x)+f(x)=0;
    if x=0 ->
    f(-0)+f(0)=0;
    2*f(0)=0;
    f(0)=0;
    But, the function should be defined(?) at 0. Don't know what to do with 1/x, log(x), ...
     
  8. Ratch

    New Member

    Mar 20, 2007
    1,068
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    Asad1,

    Even functions are mirror images with respect to only the y-axis. Odd junctions are upside mirror images of with respect to both the x and y axis. Therefore odd functions reference both the x and y axis, which intersect at the origin. It is wrong for you say "start" at zero when you mean referenced at the origin. Functions start from the minus x-axis and end at plus x-axis. Furthermore, you are asking to prove a definition, Definitions cannot be proven.

    Ratch
     
  9. Asad1

    Thread Starter Member

    Feb 11, 2009
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    Thank you soooooooooooooo much:)
     
  10. Alexei Smirnov

    Active Member

    Jan 7, 2009
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    Wait a second, being pedantic...definition of odd function is
    f(-x) = -f(x).
    This does not necessarily mean f(0) = 0, so it can (must) be proven...
     
  11. Ratch

    New Member

    Mar 20, 2007
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    Alexei Smirnov,

    The definition of a odd function does not have to be proven. There are odd functions such as the one I submitted earlier f(x)=1/x which do not pass through the origin. The plot follows its reflection with both the y and the x axis.

    Ratch
     
  12. studiot

    AAC Fanatic!

    Nov 9, 2007
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    In general Ratch has it correct.

    There is no requirement for f(0) = 0.

    You simply have to lift any even or odd function by adding a constant eg f(x) =sin(x) +1.

    There are other pitfalls though so I have appended some sketches based on straight lines.

    Many textbooks state that you can always write any function as the sum of an even plus an odd function and great use is made of this in electrical engineering. However this statement is only strictly true of certain classes of function, in particular polynomials and trigonometric functions.

    I have also displayed one very important function which is both odd and even.
     
    Last edited: Feb 12, 2009
  13. Alexei Smirnov

    Active Member

    Jan 7, 2009
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    Do you mean that not any function can be represented as a sum of even and odd functions? I think that's not true, and can easily be proven.
     
  14. studiot

    AAC Fanatic!

    Nov 9, 2007
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    Consider

    f(x) =
    \frac{1}{2}[f(x) + f(-x)] + \frac{1}{2}[f(x) - f(-x)]

    an even function plus an odd function.
     
  15. Alexei Smirnov

    Active Member

    Jan 7, 2009
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    Exactly.

    Did I misunderstand your statement:

    , in other words "not ALL functions can be represented by sum of odd and even?
     
  16. studiot

    AAC Fanatic!

    Nov 9, 2007
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    No you did not misunderstand.

    I gave you a way of deconstructing f(x) into odd and even components, so long as f(-x) is defined.
    Of course f(-x) is not defined for all functions, eg square root.
     
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