Real Rational Functions

Definition

A real rational function (or rational function or fractional function) is a function $f: \R \to \R $ for which there exists polynomials $p: \R \to \R $ and $q: \R \to \R $ so that $f(x) = p(x) / q(x)$ for all $x \in \R $.

In this case we call $p$ the numerator polynomial (and $p$ the numerator function) and $q$ the denominator polynomial (and $q$ the denominator function). Of course, the language rational is in reference to the fact that if $p$ and $q$ are integer-valued functions, then the function $f$ is a rational-valued function (seeRational Numbers).