Editing Rational function (section)

{{Short description|Ratio of polynomial functions}}
{{About||the use in automata theory|Finite-state transducer|the use in monoid theory|Rational function (monoid)}}
{{Use American English|date = January 2019}}
{{More footnotes needed|date=September 2015}}
In [[mathematics]], a '''rational function''' is any [[function (mathematics)|function]] that can be defined by a '''rational fraction''', which is an [[algebraic fraction]] such that both the [[numerator]] and the [[denominator]] are [[polynomial]]s. The [[coefficient]]s of the polynomials need not be [[rational number]]s; they may be taken in any [[field (mathematics)|field]] {{mvar|K}}. In this case, one speaks of a rational function and a rational fraction ''over {{mvar|K}}''. The values of the [[variable (mathematics)|variable]]s may be taken in any field {{mvar|L}} containing {{mvar|K}}. Then the [[domain (function)|domain]] of the function is the set of the values of the variables for which the denominator is not zero, and the [[codomain]] is {{mvar|L}}.

The set of rational functions over a field {{mvar|K}} is a field, the [[field of fractions]] of the [[ring (mathematics)|ring]] of the [[polynomial function]]s over {{mvar|K}}.