I agree with both of these examples. And here is my point. The sort of practical applications of rational functions involve very simple rational functions. S

I agree with both of these examples. And here is my point. The sort of practical applications of rational functions involve very simple rational functions. So if our motivation comes solely from word problems then the natural question is why are we learning to manipulate ( x^2 + 2xy + y^2)/(x - y) + 1/(x^2 - y^2) Has there ever occurred a practical application in which this expression has come up? If our motivation is expanding our understanding of manipulating algebraic objects then this expression is very much a practical application. It's an extension of an existing idea - rational numbers - to a new class of objects. Be able to do this sort of abstraction is useful in mathematics, and in fact is the essence of mathematical thought. It's also useful in things like programming.