algebra-precalculusfunctionsreal-analysis
I am looking for a function f(x) with a value range of [0,1].
f(x) should increase from 0 to 1 while its parameter x increases from 0 to +infinity.
f(x) increases very fast when x is small, and then very slow and eventually approach 1 when x is infinity.
Here is a figure. The green curve is what I am looking for:
Thanks.
It would be great if I can adjust the slope of the increase. Although this is not a compulsory requirement.
Here is a completely different approach that leads to much nicer behavior at the corners.
The idea is to take a line of negative unit slope, $y=\beta-x$, extending to infinity in both directions, and then squeeze it into your target rectangle using the logistic function and its inverse the logit.
The logistic function and logit contains exponentials and logarithm, but these mostly cancel out each other when we put the whole thing together, and we get $$y = \frac{y_0}{1+e^{-\beta}\frac{x}{x_0-x}}$$ We then have to decide how $\beta$ must depend on $a$ -- your desired behavior will result if we set $e^{-\beta}=(1-a)/a$, to get the final definition $$ y = \frac{y_0}{1+\frac{(1-a)x}{a(x_0-x)}}$$ This has a number of nice properties:
The first candidate that comes to mind is the function of the form
\begin{align} y(x)&=(1-x^n)^{1/n} . \end{align}
This is how the graph of this function looks for $n=5\dots12$:
Best Answer
I think this should work well for your purposes: $$ f(x) = \frac{x}{x + a} $$ Where $a$ can be any number bigger than $0$. The smaller $a$ is, the sharper the increase will be.
ADDENDUM: if you want to extend this to an odd (and continuously differentiable) function, simply take $$ f(x) = \frac{x}{|x| + a} $$