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Symmetric functions/odd even or neither

  1. Dec 14, 2008 #1
    1. The problem statement, all variables and given/known data
    I am supposed to find out if this function is symmetric with reference to the y-axis or reference to the origin.
    The function is
    F(x)=4x^2/(x^3+x)

    2. Relevant equations

    These are how to know if even or odd
    A(-x)^even power = ax^even power
    A(-x)^odd power = -ax^odd power

    3. The attempt at a solution
    I think that to solve these u change x to -x in the function right? So I got...
    F(-x)=4-x^2/(-x^3-x)
    To
    F(-x)=4+x/(-x-x)
    To
    F(-x)=4+x/-2x
    To
    F(-x)=2+x/-x
    To
    -2

    But now what, how do I tell if it's odd even or neither?? Did I even do the math right?
     
    Last edited: Dec 14, 2008
  2. jcsd
  3. Dec 14, 2008 #2

    Mark44

    Staff: Mentor

    These equations aren't what you use to determine the evenness or oddness of a function.
    If f(-x) = f(x) for all x in the domain of f, f is an even function.
    If f(-x) = -f(x) for all x in the domain of f, f is an odd function.
    No, you need some parentheses here.
    F(-x) = 4(-x)^2/((-x)^3 + (-x))
    = 4x^2/(-x^3 - x)
    = -(4x^2)/(x^3 + x)
    = ?
    What are you doing here? You can't get the first of the four lines from what you started with, and the third line doesn't follow from the second, and the fourth doesn't follow from the third.
     
  4. Dec 14, 2008 #3
    In general, a function is Even if
    [tex] F(x) = F(-x) [/tex]
    and a function is Odd if
    [tex] F(x) = -F(-x) [/tex]

    So consider in your case
    [tex] F(x) = -\frac{x^2}{x^3+x} [/tex]

    Look now at [tex] F(-x) [/tex]:

    [tex] F(-x) = -\frac{(-x)^2}{(-x)^3+(-x)} [/tex]
    [tex] = -\frac{x^2}{-x^3-x} [/tex]
    [tex] = \frac{x^2}{x^3+x} [/tex]
    [tex] = -F(x) [/tex]

    So your function is Odd.

    Another approach is to treat it as a product or quotient of even or odd functions

    Odd function * Odd function = Even function
    Even * Even = Even
    Even * Odd = Odd
    Odd * Even = Odd

    Likewise
    Odd / Odd = Even
    Even / Even = Even
    Even / Odd = Odd (this is your case)
    Odd / Even = Odd
     
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