rational function

A rational function is a function whose rule is a rational expression: it takes an input x and returns the ratio of two polynomials, f(x) = P(x)/Q(x). The simplest one, f(x) = 1/x, you have probably seen — it is the curve that races up near the y-axis and flattens out far to the sides.

Its domain is every real number except where the denominator Q(x) equals zero; those forbidden inputs are the excluded values, and the graph never has a point above them. At such a value the graph often shoots off toward infinity, producing a vertical asymptote — an invisible wall the curve approaches but never crosses.

Rational functions are the natural home for everything in this field. Simplifying corresponds to cleaning the rule, restrictions correspond to holes and asymptotes in the graph, and inverse variation y = k/x is just one famous rational function whose graph is a hyperbola. Polynomials are the special case where the denominator is a nonzero constant.