Rational function: Difference between revisions

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{{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''}}.

==Definitions==

A function <math>f(x)</math> is called a rational function if and only if it can be written in the form

:<math> f(x) = \frac{P(x)}{Q(x)} </math>

where <math>P\,</math> and <math>Q\,</math> are [[polynomial function]]s of <math>x\,</math> and <math>Q\,</math> is not the [[zero function]]. The [[___domain of a function|___domain]] of <math>f\,</math> is the set of all values of <math>x\,</math> for which the denominator <math>Q(x)\,</math> is not zero.

However, if <math>\textstyle P</math> and <math>\textstyle Q</math> have a non-constant [[polynomial greatest common divisor]] <math>\textstyle R</math>, then setting <math>\textstyle P=P_1R</math> and <math>\textstyle Q=Q_1R</math> produces a rational function

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:<math> f_1(x) = \frac{P_1(x)}{Q_1(x)}, </math>

which may have a larger ___domain than <math> f(x)</math>, and is equal to <math> f(x)</math> on the ___domain of <math> f(x).</math> It is a common usage to identify <math> f(x)</math> and <math> f_1(x)</math>, that is to extend "by continuity" the ___domain of <math> f(x)</math> to that of <math> f_1(x).</math> Indeed, one can define a rational fraction as an [[equivalence class]] of fractions of polynomials, where two fractions <math>\textstyle \frac{A(x)}{B(x)}</math> and <math>\textstyle \frac{C(x)}{D(x)}</math> are considered equivalent if <math>A(x)D(x)=B(x)C(x)</math>. In this case <math>\textstyle \frac{P(x)}{Q(x)}</math> is equivalent to <math>\textstyle \frac{P_1(x)}{Q_1(x)}.</math>.

A '''proper rational function''' is a rational function in which the [[Degree of a polynomial|degree]] of <math>P(x)</math> is less than the degree of <math>Q(x)</math> and both are [[real polynomial]]s, named by analogy to a [[fraction#Proper and improper fractions|proper fraction]] in <math>\mathbb{Q}.</math>.<ref>{{multiref|{{cite book |first1=Martin J. |last1=Corless |first2=Art |last2=Frazho |title=Linear Systems and Control |page=163 |publisher=CRC Press |date=2003 |isbn=0203911377}}|{{cite book |first1=Malcolm W. |last1=Pownall |title=Functions and Graphs: Calculus Preparatory Mathematics |page=203 |publisher=Prentice-Hall |date=1983 |isbn=0133323048}}}}</ref>

===Degree===

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which is undefined.

A [[constant function]] such as {{math|''f''(''x'') {{=}} π}} is a rational function since constants are polynomials. The function itself is rational, even though the [[value (mathematics)|value]] of {{math|''f''(''x'')}} is irrational for all ''{{mvar|x''}}.

Every [[polynomial function]] <math>f(x) = P(x)</math> is a rational function with <math>Q(x) = 1.</math> A function that cannot be written in this form, such as <math>f(x) = \sin(x),</math> is not a rational function. However, the adjective "irrational" is '''not''' generally used for functions.