For an alternative approach, if you treat this function as a time series where x is time, you can get a reasonably good approximation by performing SVD of the trajectory matrix and building the forecast from the principal components (eigen vectors) using a recurrent formula.
Here's an example:
https://apps.axibase.com/chartlab/9922f98f
* Chart 1. Function value for x in [0, 1).
* Chart 2. Function value for x in [0, 2).
* Chart 3. Function value for x in [0, 1) and extrapolated values for x in [1, 2).