How to create synthetic data for a decaying curve in order to extrapolate it beyond some point?

Question

In the following curve , I would like to extend the measurements beyond $x$=1 in order to have a better estimate of the green curve compared to red line.

Note: I do not have the analytical form of the function but only ($x$, $y$) data sets in the range (0, 1).

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Additional Question: What is the functional form of the following curve?

Answer 1

One kind of function you may try to fit this is difference of exponential function, these two timescales controls the rising and decaying timescale of your data. $$ f(t)=A\left(\exp(-t/\tau_1)-\exp(-t/\tau_2)\right) \theta(t)\\ $$ $H(t)$ is the Heaviside step function $\theta(t)=1$ only when $t\geq 0$.

A plot using Wolfram Alpha $(\exp(-t/3)-\exp(-t/1)) \theta(t)$

Scipy's curve_fit could be used to optimize these parameters.