Figuring Out Function from Curve: Is it Possible?

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Given a function, it is easy enough to plot its curve, just by substituting numerical values.

But is the reverse possible? I mean, if you're given a curve's figure, can you figure out the function that represents it (provided that the curve is not a well-known one like a parabola or ellipse) ?
 

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There is a whole branch of mathematics devoted to curve fitting. Depending on how precise you want your curve to fit, you can end up with a simple polynomial or something very complicated. Sometimes it would be absurd to have the curve go exactly through every data point on your plotted data, especially when it's clear that one or more of your data points involve an erroneous reading or the reading errors in most measurements are significant.

Curve-fitting also offers the opportunity to replace a complicated function with a much simpler function but still a very close approximation, over a region of interest, to make calculations easier or more efficient.

Any particular plot of data can be fitted by an almost infinite number of curves, so there is plenty of scope for originality!