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The typical rational function has the form p(x)/q(x) where p and q are polynomials. p(x) is called the numerator and q(x) is called the denominator. Example: In the rational function. which simplifies to simplifies to (x+2)/(x-3), but the x = 2 is NOT in the domain of g(x) whereas it IS in the domain of (x+2)/(x-3).
The rational function is called the improper rational function, if the degree of denominator $H(x)$ is less than or equal to the degree of numerator $G(x)$. With the help of division method, we can express an improper rational function as a sum of a polynomial and a proper rational function.