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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</math>, and is equal to <math>f</math> on the ___domain of <math>f.</math> It is a common usage to identify <math> f</math> and <math> f_1</math>, that is to extend "by continuity" the ___domain of <math>f</math> to that of <math>f_1.</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>frfrt
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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>
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