How to model positive S-shaped-function? I need to transform my data into a function shown below. 
My data should fit in the range from 0 to 1. The inflection point should be on 0.5. 
How I can do this mathematically? 
Is there any similar function to model data like this?

 A: The sigmoid, S-shaped or ogive curve shown in your plot is ubiquitous in nature. Geoffrey West's recent book Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies goes into great depth in elucidating that. Another excellent resource is D'Arcy Thompson's On Growth and Form, much older but still a classic. The bottom line is that symmetric S-shapes are described using Gaussian assumptions. 
There are many approaches to modeling these functions beginning with the Gompertz curve--the first model truly nonlinear in the parameters (@1820). Of course, 100 years earlier, Riccati introduced quadratics into differential equations but his "nonlinearity" was in the inputs, not the parameters. About 20 years after Gompertz, Verhulst developed the classic logistic curve, the assumption underlying today's logistic regression. Gompertz and logistic curves differ nontrivially in that the Gompertz is asymmetric and long-tailed while the logistic is symmetric and short-tailed. Given the difficulties many analysts experience in trying to fit rare or large magnitude (+/-) events based on logistic regression, one wishes software developers would come up with canned tools for Gompertz regression.
Since Gompertz and Verhulst, there have been many more S-shaped growth curve functions: for instance, the Pearl curve, the Fisher-Pry transformation, logit functions also do this and more. Wrt model building, multiplicative, log-log models reproduce S-shapes. Introducing quadratic (squared) polynomials into a regression model will work as well. This wiki article goes into much greater depth about sigmoid functions and their variant specifications... https://en.wikipedia.org/wiki/Sigmoid_function.
With all due respect to Geoffrey West, that S-shaped curves are ubiquitous does not qualify them as universal laws. Equally ubiquitous phenomena are extreme value distributed information or data in which large magnitude (+/-) events overwhelm normal, bell-shaped, ordinary information. 
I don't know LaTeX or the LaTeX-like functions used on CV but this wiki piece (https://en.wikipedia.org/wiki/Logit, see the Definition section) shows how to fit an S-shaped, logistic curve bounded between 0 and 1 for any number a using an inverse logit function. In words, the inverse logit is exp(a)/(exp(a)+1).
A: You can use the inverse logit
x <- rnorm(100, 0, 5)

inv_logit <- function(x) {
  return(1 / (1 + exp(- x)))
}

y <- inv_logit(x)

plot(y ~ x)

