Rational function interpolation

My first thought was this was kind of obvious but apparently when people want to find what this finds the answer to they use something called http://en.wikipedia.org/wiki/Pad%C3%A9_approximation. This is a much simpler way that I thought of.

Say you have your unknown function:

The four fractions at the bottom are what the function equals at x=1,2,3,4, you'll notice if you use regular interpolation with these numbers it won't even be close to the unknown function above.

But what you can try is to interpolate the top half of each fraction and bottom half separately, like so (Make sure they are in lowest terms or this won't work):

There that matches the unknown function very well...