I'm looking for a function that increases slowly at first and then moderately so, between 0 and 1, for x starting from 1 with no fixed upper limit. Something like the chart below. It would be great to be able to control the slope.
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Take the log of your function. If it looks like a straight line, exponential would be a good fit. If it looks like the log function (increasing monotonically, but growth slowing down), polynomial would be a better fit. See discussion on exponential and polynomial below. – Godfather Sep 10 '21 at 14:40
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Look at the family $f_n(x) = x^n$ which always is 0 at 0 and 1 at 1... If you have no fixed upper limit, try $f_n(x) = Ax^n$ for another slope-control parameter.
gt6989b
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You can also look at $f(x) = \exp(\alpha x) - 1$ for $\alpha > 0$ and use $\alpha$ to control the slope.
Of course $f_n(x) = x^n$ is also a great choice as the previous poster suggested.
I would suggest constraining the problem a bit more to help find the right family of functions to fit your data.
Godfather
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1May be useful for OP to note that the first function you mentioned is an exponential function while the second one is polynomial. Both are commonly used to fit to data. – Sep 10 '21 at 03:09