Candidate Functions f(x) for Domain R, Image (-1,1)

[longrob]

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]

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

 

[longrob]

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

 

[trambolin]

check sigmoid curves

 

[longrob]

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]

Well, you might try with something like:

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

 

[g_edgar]

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: (arctan(x))^{1/3}

 

[longrob]

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: (arctan(x))^{1/3}

Perfect. Thank you.