# Rational function

**Rational function** is a quotient of two polynomial functions. It distinguishes from *irrational function* which cannot be written as a ratio of two polynomials.

## Definition

A rational function is a function of the form

where *s* and *t* are polynomial function in *x* and *t* is not the zero polynomial. The domain of *f* is the set of all points *x* for which the denominator *t*(*x*) is not zero.

On the graph restricted values of an axis form a straight line, called asymptote, which cannot be crossed by the function. If zeros of numerator and denominator are equal, then the function is a horizontal line with the hole on a restricted value of *x*.

## Examples

Let's see an example of in a factored form: . Obviously, roots of denominator is -5 and 4. That is, if *x* takes one of these two values, the denominator becomes equal to zero. Since the division by zero is impossible, the function is not defined or discontinuous at *x* = -5 and *x* = 4.

The function is continuous at all other values for *x*. The domain (area of acceptable values) for the function, as expressed in interval notation, is: