Graphs of Rational Functions - Revision history

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Definitions

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← Older revision		Revision as of 19:48, 21 October 2021
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	*Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007), "Section 3.4. Rational Function Interpolation and Extrapolation", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8		*Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007), "Section 3.4. Rational Function Interpolation and Extrapolation", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
	*[https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/graph-rational-functions/ Graph Rational Functions], Lumen Learning		*[https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/graph-rational-functions/ Graph Rational Functions], Lumen Learning
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikipedia.org/wiki/Rational_function Rational function, Wikipedia] under a CC BY-SA license
		+	* [https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/graph-rational-functions/ Graph Rational Functions, Lumen Learning College Algebra] under a CC BY license

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 ==References==
-{{Reflist}}
 *Martin J. Corless, Art Frazho, Linear Systems and Control, p. 163, CRC Press, 2003 ISBN 0203911377.
 *Malcolm W. Pownall, Functions and Graphs: Calculus Preparatory Mathematics, p. 203, Prentice-Hall, 1983 ISBN 0133323048.

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 where <math>P\,</math> and <math>Q\,</math> are polynomial functions of <math>x\,</math> and <math>Q\,</math> is not the zero function.  The domain of f 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
+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
 :<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>\frac{A(x)}{B(x)}</math> and <math>\frac{C(x)}{D(x)}</math> are considered equivalent if <math>A(x)D(x)=B(x)C(x)</math>. In this case <math>\frac{P(x)}{Q(x)}</math> is equivalent to <math>\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>\frac{A(x)}{B(x)}</math> and <math>\frac{C(x)}{D(x)}</math> are considered equivalent if <math>A(x)D(x)=B(x)C(x)</math>. In this case <math>\frac{P(x)}{Q(x)}</math> is equivalent to <math>\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 no greater than the degree of <math>Q(x)</math> and both are real polynomials.

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 *"Rational function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 *Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007), "Section 3.4. Rational Function Interpolation and Extrapolation", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
-*[https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/graph-rational-functions/], Lumen Learning
+*[https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/graph-rational-functions/ Graph Rational Functions], Lumen Learning

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-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]] ''K''. In this case, one speaks of a rational function and a rational fraction ''over K''. The values of the [[variable (mathematics)|variable]]s may be taken in any field ''L'' containing ''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 ''L''.
+In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is L.
-The set of rational functions over a field ''K'' is a field, the [[field of fractions]] of the [[ring (mathematics)|ring]] of the [[polynomial function]]s over ''K''.
+The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K.
 ==Definitions==

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 ==Examples==
 [[File:RationalDegree3.svg|thumb|Rational function of degree 3, with a graph of degree 3: <math>y = \frac{x^3-2x}{2(x^2-5)}</math>]]
-[[File:RationalDegree2byXedi.svg|thumb|Rational function of degree 2, with a graph of degree 3: {\displaystyle y={\frac {x^{2}-3x-2}{x^{2}-4}}}y={\frac {x^{2}-3x-2}{x^{2}-4}}]]
+[[File:RationalDegree2byXedi.svg|thumb|Rational function of degree 2, with a graph of degree 3: <math>y = \frac{x^2-3x-2}{x^2-4}</math>]]
 The rational function

@@ Line 36: / Line 36: @@
 ==Examples==
-{{multiple image
+[[File:RationalDegree3.svg|thumb|Rational function of degree 3, with a graph of degree 3: {\displaystyle y={\frac {x^{3}-2x}{2(x^{2}-5)}}}y={\frac {x^{3}-2x}{2(x^{2}-5)}}]]
- | header = Examples of rational functions
+[[File:RationalDegree2byXedi.svg|thumb|Rational function of degree 2, with a graph of degree 3: {\displaystyle y={\frac {x^{2}-3x-2}{x^{2}-4}}}y={\frac {x^{2}-3x-2}{x^{2}-4}}]]
 The rational function

← Older revision		Revision as of 21:31, 15 September 2021
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−	~~{{About\|\|the use in automata theory\|Finite-state transducer\|the use in monoid theory\|Rational function (monoid)}}~~
−	~~{{Use American English\|date = January 2019}}~~
−	~~{{Short description\|Ratio of polynomial functions}}~~
−	~~{{More footnotes\|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]] ''K''. In this case, one speaks of a rational function and a rational fraction ''over K''. The values of the [[variable (mathematics)\|variable]]s may be taken in any field ''L'' containing ''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 ''L''.		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]] ''K''. In this case, one speaks of a rational function and a rational fraction ''over K''. The values of the [[variable (mathematics)\|variable]]s may be taken in any field ''L'' containing ''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 ''L''.