what's this function ?

**longrob** · Jun27-09, 12:33 AM

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

Thanks
LR

**snipez90** · Jun27-09, 01:49 AM

Well the horizontal asymptotes are at 1 and - 1, so a suitable rational function should do the trick.

**longrob** · Jun27-09, 04:23 AM

Thanks, but I've not been able to find one. Any suggestions ?

**trambolin** · Jun27-09, 04:35 AM

check sigmoid curves

**longrob** · Jun27-09, 05:04 AM

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.

**arildno** · Jun27-09, 05:42 AM

Well, you might try with something like:

x>0: $LaTeX Code: f(x)=C\\sqrt{arctan(|x|)}$
x<0: $LaTeX Code: f(x)=-C\\sqrt{arctan(|x|)}, C=\\sqrt{\\frac{2}{\\pi}}$

**g_edgar** · Jun27-09, 07:08 AM

Originally Posted by snipez90

Well the horizontal asymptotes are at 1 and - 1, so a suitable rational function should do the trick.

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: $LaTeX Code: (arctan(x))^{1/3}$

**longrob** · Jun28-09, 07:37 PM

Originally Posted by g_edgar

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: $LaTeX Code: (arctan(x))^{1/3}$

Perfect. Thank you.