Rational function

From Citizendium, the Citizens' Compendium
Jump to: navigation, search
This article is a stub and thus not approved.
Main Article
Talk
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.

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: