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I need analytical formula of two funcitons. On picture, I painted plot, how functions should look like. enter image description here

The left tail of both function does not matter, ie behavior of functions on $(0, -\inf)$ does not matter. I need both functions to be asymptotic to 0.5, one decreaxing from [0,1], second increacing from [0,0]. I prefer one parameter of each function to control the rate of decrase.

I played with transformations of $1/x$, but with no luck. My unrealistic idea of red curve is something like $exp(-x + log(-0.5)) + 0.5$ but obviously it is not solution.

Thx for any advices. Also any suggestions and hints are very welcomed.

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    This question is very vague, i.e. it is unclear how the function should exatly look like (so it is very vague to give an analytical expression for it). From which problem does this arise? – 2017-02-06
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    Sorry, Ill try to edit it and make problem more clear. – 2017-02-06

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For the red function I would use exponential decay $ f(x) = \frac{1 + e^{-\lambda x}}{2}$ while for the blue function I would use the red function reflected along $y=0.5$ so $ g(x) = \frac{1 - e^{-\lambda x}}{2}$ in both cases $\lambda$ is the value you adjust to get different decay rates.