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Do you think that it is possible to find a $f$ and a $g$ such that $\forall x \gt 1, \forall y \gt 1$ then

$$f(x) \gt 0,$$ $$g(x) \gt 0,$$ $$f(x) \neq x,$$ $$g(x) \neq x,$$

and

$$f\left({1\over 1-{1\over x}-{1\over y}}\right) = {1\over 1-{1\over g(x)}-{1\over g(y)}}$$

and

$$g\left({1\over 1-{1\over x}-{1\over y}}\right) = {1\over 1-{1\over f(x)}-{1\over f(y)}}$$

Like suggested to avoid division by 0, I restrict the domain of the function at the cases where : $${1\over f(x)}+{1\over f(y)} \lt 1$$ $${1\over g(x)}+{1\over g(y)} \lt 1$$

(I know that ${1\over x}+{1\over y} \lt 1$ because I know (given the characteritics) that one function will give an output greater than its parameter and the other will give an output lower)

This is a more restrictive version of my other question : Find a possible f(x) and g(x)

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infiniteLoop
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    Just out of curiosity: why do you need such a pair of functions? – 5xum Feb 19 '16 at 07:39
  • What are the domains of $f,g$? – Hagen von Eitzen Feb 19 '16 at 07:41
  • @HagenvonEitzen : They take reals and give reals – infiniteLoop Feb 19 '16 at 07:43
  • So if there exists $x>1$ with $f(x)=3$, for example, then there is no $y>1$ with $f(y)=\frac32$ except possibly $y=\frac1{1-\frac1x}=1+\frac1{x-1}$? – Hagen von Eitzen Feb 19 '16 at 07:46
  • The equations you have, shouldn't they be defined only for $x,y>2$ (for $x=y=2$ we are looking at $f(\frac{1}{0})$)? – Clément Guérin Feb 19 '16 at 08:07
  • @HagenvonEitzen : You are talking about a division by zero like Clément wright ? – infiniteLoop Feb 19 '16 at 08:19
  • @ClémentGuérin : Yes you are wright I should restrict the domain to avoid this case of division by zero. I could say $x, y > 2$ but I know that one function will give an output greater than its parameter and the other will give an output lower. So I imagine that is not sufficient – infiniteLoop Feb 19 '16 at 08:25
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    @5xum : This is for a zero-sum game problem, I would like to know if it's possible to make a translation of a problem into one with less or more payouts (just translating the prices for the outcomes) but having the same other characteristics – infiniteLoop Feb 19 '16 at 08:31

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