Rational Functions

Rational Functions are any functions which can be defined by a rational fraction for example an algebraic fraction such that both the numerator and the denominator are polynomials. A function  is called a rational function if and only if it can be written in the form

. Where  and  are polynomials in X and Q is not the zero polynomial. The domain of f is the set of all points X for which the denominator  is not zero. However, if P and Q have a non constant polynomial greatest common divisor R, then setting  and  produces a rational function . The degree of a rational function is the maximum of the degrees of its constituent polynomials P and Q. Rational functions with degree 1 are called Mobius transformations and form the automorphisms group of the Riemann Sphere. Rational functions are representative examples of meromorphic functions. 

Graph of a rational function: