From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: looking for functions
Date: 4 Feb 1997 20:48:52 GMT

In article <32F794B7.3E72@ltsi4.univ-rennes1.fr>,
ayad  <ayad@ltsi4.univ-rennes1.fr> wrote:

>I am looking for functions wich verifies the folowing conditions :
>
>1)  x>0
>2)  0<f(x)<1
>3)   f(x)+f(1/x)=1

I will tell you what _all_ of them are; you decide which ones you want.
Simply let  h  be any function on  (0, oo)  which is everywhere properly
bounded by  1  (e.g.  h(x) =  exp(-x) or sin(x)/2 ), and let
	f(x) =  ( h(x) - h(1/x) + 2 )/4.
(If you have a particular  f  in mind, you can make it appear in this
formula by simply taking  h(x) = 2 f(x) - 1.)

The only polynomial  f  meeting the given conditions is constant. One
_can_ find rational functions  f; the simplest, I suppose, is 
f(x) = x/(x+1).

dave