Is there a general logit -type function (or another sigmodal function with similar shape) which returns f(x) between 0 and 1?

Question

I'm looking for a general function that can fit experimental data that has a profile/trend similar to the logit function $f(x) = \log x/(1-x)$, but where all f(x) values are between 0 and 1.

Is there a generalised version of the logit function, or any other transformed sigmoidal function, that I can scale and fit my data below?

f(0.005) = 0.005

f(0.04) = 0.06

f(0.1) = 0.22

f(0.2) = 0.37

f(0.3) = 0.47

f(0.4) = 0.48

f(0.5) = 0.53

f(0.6) = 0.55

f(0.7) = 0.56

f(0.8) = 0.58

f(0.9) = 0.66

f(0.96) = 0.79

f(0.999) = 1.00

I'm also attaching an image of the plot. Thanks!

curve_fit_problem

Answer 1

What about this one? $$f(x)=4(x-0.5)^3+0.5$$

There is no logarithm or exponential, but the shape of the curve is very similar.

Answer 2

It's quite a good fit to $$ x + .684 x (1-x)+8.98 x^2 (1-x)-48.6 x^3 (1-x)+79.3 x^4 (1-x)-45.7 x^5 (1-x) $$ obtained by linear least squares.