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What's this function ?

  1. Jun 26, 2009 #1
    Hi all

    What are some candidate functions f(x) that satisfy these conditions:
    1. domain of f is R
    2. image of f is (-1,1)
    2. Smooth and continuous everywhere
    3. first derivative undefined at x=0
    4. f(x)-->1 as x--> inf
    5. f(x)-->-1 as x--> -inf

  2. jcsd
  3. Jun 27, 2009 #2
    Well the horizontal asymptotes are at 1 and - 1, so a suitable rational function should do the trick.
  4. Jun 27, 2009 #3
    Thanks, but I've not been able to find one. Any suggestions ?
  5. Jun 27, 2009 #4
    check sigmoid curves
  6. Jun 27, 2009 #5
    Thanks. I considered arctan already, but since this function goes momentarily vertical zero arctan doesn't work. Same with a Gompertz function and Richards curve (I think). Also, this function appears to be odd, so that would rule out a Gompterz function also. Are there Sigmoid curves that are odd ? I don't know much about them, except in population models, and in those models a disappearing first derivative isn't too desirable I guess.
  7. Jun 27, 2009 #6


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    Well, you might try with something like:

    x>0: [itex]f(x)=C\sqrt{arctan(|x|)}[/itex]
    x<0: [itex]f(x)=-C\sqrt{arctan(|x|)}, C=\sqrt{\frac{2}{\pi}}[/itex]
  8. Jun 27, 2009 #7
    No, a rational function has the same asymptote at both ends.

    So: in rejecting arctan, you say that you WANT it to be vertical at zero? (arctan has slope 1 at zero). Then take the cube root: [tex](arctan(x))^{1/3}[/tex]

    Attached Files:

  9. Jun 28, 2009 #8
    Perfect. Thank you.
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