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Even or odd?

  1. Dec 8, 2004 #1
    I just can't seem to grasp this! I have no problems finding out if a function let's say [tex]x-2x^2[/tex] is an even or odd function, but when the function is defined differently along different part along the x-axis then I don't understand anything! This function:
    [tex]f(x)=\left\{\begin{array}{cc}0 &\mbox{ if }
    -2\leq x<0\\(1/2)x & \mbox{ if }0\leq x<2\end{array}\right[/tex]

    Someone help me please! :cry:
    This function is supposed to be neither acutally, but I have no idea how to show this...
     
  2. jcsd
  3. Dec 8, 2004 #2

    arildno

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    Well let x=1
    Then f(1)=1/2, but f(-1)=0 which is not equal to either 1/2 or -1/2.
    Hence, f(x) is neither even nor odd.
     
  4. Dec 8, 2004 #3

    Gokul43201

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    Extra Note : Keep in mind that the above method of comparing f(a) with f(-a) for a particular choice of 'a', can be used only to show that f is neither even nor odd.

    To show that some f is even or odd in a given domain, you must show that the relevant relationship holds for all 'a' in the specified domain.
     
  5. Dec 8, 2004 #4

    arildno

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    As Gokul said, I gave a SUFFICIENT proof of f being neither even or odd, by providing a COUNTER-EXAMPLE (of even-ness and odd-ness).
     
  6. Dec 8, 2004 #5
    As Gokul said, it only gives me the answer at the point a. I can show that each of the functions separately are either even or odd (or neither), but how do I show this for a given domain...? I know the definition for an odd function is f(-x) = -f(x), and for an even function f(-x) = f(x), but in what function should I put in the negative x?? I have two (sometimes more) to chose from, 0 and (1/2)x. Help! Nå ser jeg jo at jeg sikkert kunne skrevet norsk her også...
     
  7. Dec 8, 2004 #6

    arildno

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    It's enough with a single counter-example to prove that it is neither even or odd on the given domain (the condition for even-ness must hold for ALL members in the domain in order for the function to be even).
     
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