Difference between revisions of "Graphs of Rational Functions"

Revision as of 20:16, 15 September 2021

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 of the polynomial functions over K.

Definitions

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

f(x)={\frac {P(x)}{Q(x)}}

where $P\,$ and $Q\,$ are polynomial functions of $x\,$ and $Q\,$ is not the zero function. The domain of f is the set of all values of $x\,$ for which the denominator $Q(x)\,$ is not zero.

However, if $\textstyle P$ and $\textstyle Q$ have a non-constant polynomial greatest common divisor $\textstyle R$ , then setting $\textstyle P=P_{1}R$ and $\textstyle Q=Q_{1}R$ produces a rational function

f_{1}(x)={\frac {P_{1}(x)}{Q_{1}(x)}},

which may have a larger domain than $f(x)$ , and is equal to $f(x)$ on the domain of $f(x).$ It is a common usage to identify $f(x)$ and $f_{1}(x)$ , that is to extend "by continuity" the domain of $f(x)$ to that of $f_{1}(x).$ Indeed, one can define a rational fraction as an equivalence class of fractions of polynomials, where two fractions ${\frac {A(x)}{B(x)}}$ and ${\frac {C(x)}{D(x)}}$ are considered equivalent if $A(x)D(x)=B(x)C(x)$ . In this case ${\frac {P(x)}{Q(x)}}$ is equivalent to ${\frac {P_{1}(x)}{Q_{1}(x)}}$ .

A proper rational function is a rational function in which the Degree of a polynomial|degree of $P(x)$ is no greater than the degree of $Q(x)$ and both are real polynomials.

Degree

There are several non equivalent definitions of the degree of a rational function.

Most commonly, the degree of a rational function is the maximum of the degrees of its constituent polynomials P and Q, when the fraction is reduced to lowest terms. If the degree of f is d, then the equation

f(z)=w\,

has d distinct solutions in z except for certain values of w, called critical values, where two or more solutions coincide or where some solution is rejected at infinity (that is, when the degree of the equation decrease after having cleared the denominator).

In the case of complex coefficients, a rational function with degree one is a Möbius transformation.

The degree of the graph of a rational function is not the degree as defined above: it is the maximum of the degree of the numerator and one plus the degree of the denominator.

In some contexts, such as in asymptotic analysis, the degree of a rational function is the difference between the degrees of the numerator and the denominator.

In network synthesis and network analysis, a rational function of degree two (that is, the ratio of two polynomials of degree at most two) is often called a biquadratic function.

Examples

Rational function of degree 3, with a graph of degree 3:

y={\frac {x^{3}-2x}{2(x^{2}-5)}}

Rational function of degree 2, with a graph of degree 3:

y={\frac {x^{2}-3x-2}{x^{2}-4}}

The rational function

f(x)={\frac {x^{3}-2x}{2(x^{2}-5)}}

is not defined at

x^{2}=5\Leftrightarrow x=\pm {\sqrt {5}}.

It is asymptotic to ${\tfrac {x}{2}}$ as $x\to \infty .$

The rational function

f(x)={\frac {x^{2}+2}{x^{2}+1}}

is defined for all real numbers, but not for all complex numbers, since if x were a square root of -1 (i.e. the imaginary unit or its negative), then formal evaluation would lead to division by zero:

f(i)={\frac {i^{2}+2}{i^{2}+1}}={\frac {-1+2}{-1+1}}={\frac {1}{0}},

which is undefined.

A constant function such as f(x) = π is a rational function since constants are polynomials. The function itself is rational, even though the value of f(x) is irrational for all x.

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

The rational function $f(x)={\tfrac {x}{x}}$ is equal to 1 for all x except 0, where there is a removable singularity. The sum, product, or quotient (excepting division by the zero polynomial) of two rational functions is itself a rational function. However, the process of reduction to standard form may inadvertently result in the removal of such singularities unless care is taken. Using the definition of rational functions as equivalence classes gets around this, since x/x is equivalent to 1/1.

Sketch a graph of rational functions

Evaluate the function at 0 to find the y-intercept.
Factor the numerator and denominator.
For factors in the numerator not common to the denominator, determine where each factor of the numerator is zero to find the x-intercepts.
Find the multiplicities of the x-intercepts to determine the behavior of the graph at those points.
For factors in the denominator, note the multiplicities of the zeros to determine the local behavior. For those factors not common to the numerator, find the vertical asymptotes by setting those factors equal to zero and then solve.
For factors in the denominator common to factors in the numerator, find the removable discontinuities by setting those factors equal to 0 and then solve.
Compare the degrees of the numerator and the denominator to determine the horizontal or slant asymptotes.
Sketch the graph.

References

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.
Glisson, Tildon H., Introduction to Circuit Analysis and Design, Springer, 2011 ISBN ISBN 9048194431.
"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
[1], Lumen Learning

@@ Line 8: / Line 8: @@
 :<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.
+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
@@ Line 16: / Line 16: @@
 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 polynomial]]s.<ref>{{multiref|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}}.}}</ref>
+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.
 ===Degree===
 There are several non equivalent definitions of the degree of a rational function.
-Most commonly, the ''degree'' of a rational function is the maximum of the [[degree of a polynomial|degrees]] of its constituent polynomials {{math|''P''}} and {{math|''Q''}}, when the fraction is reduced to [[lowest terms]]. If the degree of {{math|''f''}} is {{math|''d''}}, then the equation
+Most commonly, the degree of a rational function is the maximum of the degrees of its constituent polynomials P and Q, when the fraction is reduced to lowest terms. If the degree of f is d, then the equation
 :<math>f(z) = w \,</math>
-has {{math|''d''}} distinct solutions in {{math|''z''}} except for certain values of {{math|''w''}}, called ''critical values'', where two or more solutions coincide or where some solution is rejected [[point at infinity|at infinity]] (that is, when the degree of the equation decrease after having [[clearing denominators|cleared the denominator]]).
+has d distinct solutions in z except for certain values of w, called critical values, where two or more solutions coincide or where some solution is rejected at infinity (that is, when the degree of the equation decrease after having cleared the denominator).
-In the case of [[complex number|complex]] coefficients, a rational function with degree one is a ''[[Möbius transformation]]''.
+In the case of complex coefficients, a rational function with degree one is a Möbius transformation.
-The [[degree of an algebraic variety|degree]] of the graph of a rational function is not the degree as defined above: it is the maximum of the degree of the numerator and one plus the degree of the denominator.
+The degree of the graph of a rational function is not the degree as defined above: it is the maximum of the degree of the numerator and one plus the degree of the denominator.
-In some contexts, such as in [[asymptotic analysis]], the ''degree'' of a rational function is the difference between the degrees of the numerator and the denominator.
+In some contexts, such as in asymptotic analysis, the degree of a rational function is the difference between the degrees of the numerator and the denominator.
-In [[network synthesis]] and [[Network analysis (electrical circuits)|network analysis]], a rational function of degree two (that is, the ratio of two polynomials of degree at most two) is often called a '''{{vanchor|biquadratic function}}'''.<ref>Glisson, Tildon H., ''Introduction to Circuit Analysis and Design'', Springer, 2011 ISBN {{ISBN|9048194431}}.</ref>
+In network synthesis and network analysis, a rational function of degree two (that is, the ratio of two polynomials of degree at most two) is often called a biquadratic function.
 ==Examples==
@@ Line 53: / Line 53: @@
 :<math>f(x) = \frac{x^2 + 2}{x^2 + 1}</math>
-is defined for all [[real number]]s, but not for all [[complex number]]s, since if ''x'' were a square root of <math>-1</math> (i.e. the [[imaginary unit]] or its negative), then formal evaluation would lead to division by zero:
+is defined for all real numbers, but not for all complex numbers, since if x were a square root of -1 (i.e. the imaginary unit or its negative), then formal evaluation would lead to division by zero:
 :<math>f(i) = \frac{i^2 + 2}{i^2 + 1} = \frac{-1 + 2}{-1 + 1} = \frac{1}{0},</math>
@@ Line 59: / Line 59: @@
 which is undefined.
-A [[constant function]] such as ''f''(''x'') = π is a rational function since constants are polynomials. The function itself is rational, even though the [[value (mathematics)|value]] of ''f''(''x'') is irrational for all ''x''.
+A constant function such as ''f''(''x'') = π is a rational function since constants are polynomials. The function itself is rational, even though the [[value (mathematics)|value]] of ''f''(''x'') is irrational for all ''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.
+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.
-The rational function <math>f(x) = \tfrac{x}{x}</math> is equal to 1 for all ''x'' except 0, where there is a [[removable singularity]]. The sum, product, or quotient (excepting division by the zero polynomial) of two rational functions is itself a rational function. However, the process of reduction to standard form may inadvertently result in the removal of such singularities unless care is taken. Using the definition of rational functions as equivalence classes gets around this, since ''x''/''x'' is equivalent to 1/1.
+The rational function <math>f(x) = \tfrac{x}{x}</math> is equal to 1 for all ''x'' except 0, where there is a removable singularity. The sum, product, or quotient (excepting division by the zero polynomial) of two rational functions is itself a rational function. However, the process of reduction to standard form may inadvertently result in the removal of such singularities unless care is taken. Using the definition of rational functions as equivalence classes gets around this, since ''x''/''x'' is equivalent to 1/1.
-==Taylor series==
+==Sketch a graph of rational functions==
-The coefficients of a [[Taylor series]] of any rational function satisfy a [[Recurrence relation|linear recurrence relation]], which can be found by equating the rational function to a Taylor series with indeterminate coefficients, and collecting [[like terms]] after clearing the denominator.
+# Evaluate the function at 0 to find the y-intercept.
+# Factor the numerator and denominator.
-For example,
+# For factors in the numerator not common to the denominator, determine where each factor of the numerator is zero to find the x-intercepts.
+# Find the multiplicities of the x-intercepts to determine the behavior of the graph at those points.
-:<math>\frac{1}{x^2 - x + 2} = \sum_{k=0}^{\infty} a_k x^k.</math>
+# For factors in the denominator, note the multiplicities of the zeros to determine the local behavior. For those factors not common to the numerator, find the vertical asymptotes by setting those factors equal to zero and then solve.
+# For factors in the denominator common to factors in the numerator, find the removable discontinuities by setting those factors equal to 0 and then solve.
-Multiplying through by the denominator and distributing,
+# Compare the degrees of the numerator and the denominator to determine the horizontal or slant asymptotes.
+# Sketch the graph.
-:<math>1 = (x^2 - x + 2) \sum_{k=0}^{\infty} a_k x^k</math>
-:<math>1 = \sum_{k=0}^{\infty} a_k x^{k+2} - \sum_{k=0}^{\infty} a_k x^{k+1} + 2\sum_{k=0}^{\infty} a_k x^k.</math>
-After adjusting the indices of the sums to get the same powers of ''x'', we get
-:<math>1 = \sum_{k=2}^{\infty} a_{k-2} x^k - \sum_{k=1}^{\infty} a_{k-1} x^k + 2\sum_{k=0}^{\infty} a_k x^k.</math>
-Combining like terms gives
-:<math>1 = 2a_0 + (2a_1 - a_0)x + \sum_{k=2}^{\infty} (a_{k-2} - a_{k-1} + 2a_k) x^k.</math>
-Since this holds true for all ''x'' in the radius of convergence of the original Taylor series, we can compute as follows.  Since the [[constant term]] on the left must equal the constant term on the right it follows that
-:<math>a_0 = \frac{1}{2}.</math>
-Then, since there are no powers of ''x'' on the left, all of the [[coefficient]]s on the right must be zero, from which it follows that
-:<math>a_1 = \frac{1}{4}</math>
-:<math>a_k = \frac{1}{2} (a_{k-1} - a_{k-2})\quad \text{for}\ k \ge 2.</math>
-Conversely, any sequence that satisfies a linear recurrence determines a rational function when used as the coefficients of a Taylor series. This is useful in solving such recurrences, since by using [[partial fraction|partial fraction decomposition]] we can write any proper rational function as a sum of factors of the form {{nowrap|1 / (''ax'' + ''b'')}} and expand these as [[geometric series]], giving an explicit formula for the Taylor coefficients; this is the method of [[generating functions]].
-==Abstract algebra and geometric notion== <!-- Rational expression redirects here -->
-In [[abstract algebra]] the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any [[field (mathematics)|field]].  In this setting given a field ''F'' and some indeterminate ''X'', a '''rational expression''' is any element of the [[field of fractions]] of the [[polynomial ring]] ''F''[''X''].  Any rational expression can be written as the quotient of two polynomials ''P''/''Q'' with ''Q'' ≠ 0, although this representation isn't unique.  ''P''/''Q'' is equivalent to ''R''/''S'', for polynomials ''P'', ''Q'', ''R'', and ''S'', when ''PS'' = ''QR''.  However, since ''F''[''X''] is a [[unique factorization domain]], there is a [[irreducible fraction|unique representation]] for any rational expression ''P''/''Q'' with ''P'' and ''Q'' polynomials of lowest degree and ''Q'' chosen to be [[monic polynomial|monic]].  This is similar to how a [[Fraction (mathematics)|fraction]] of integers can always be written uniquely in lowest terms by canceling out common factors.
-The field of rational expressions is denoted ''F''(''X''). This field is said to be generated (as a field) over ''F'' by (a [[transcendental element]]) ''X'', because ''F''(''X'') does not contain any proper subfield containing both ''F'' and the element ''X''.
-===Complex rational functions===
-<gallery caption = "Julia sets for rational maps ">
-Julia set f(z)=1 over az5+z3+bz.png| <math>\frac{1}{ az^5+z^3+bz}</math>
-Julia set f(z)=1 over z3+z*(-3-3*I).png|<math>\frac{1}{z^3+z(-3-3i)}</math>
-Julia set for f(z)=(z2+a) over (z2+b) a=-0.2+0.7i , b=0.917.png|<math>\frac{z^2 - 0.2 + 0.7i}{z^2 + 0.917}</math>
-Julia set for f(z)=z2 over (z9-z+0.025).png| <math>\frac{z^2}{z^9 - z + 0.025}</math>
-</gallery>
-In [[complex analysis]], a rational function
-:<math>f(z) = \frac{P(z)}{Q(z)}</math>
-is the ratio of two polynomials with complex coefficients, where {{math|''Q''}} is not the zero polynomial and {{math|''P''}} and {{math|''Q''}} have no common factor (this avoids {{math|''f''}} taking the indeterminate value 0/0).
-The domain of {{mvar|f}} is the set of complex numbers such that <math>Q(z)\ne 0,</math> and its range is the set of the complex numbers {{mvar|w}} such that <math>P(z)\ne wQ(z).</math>
-Every rational function can be naturally extended to a function whose domain and range are the whole [[Riemann sphere]] ([[complex projective line]]).
-Rational functions are representative examples of [[meromorphic function]]s.
-Iteration of rational functions (maps)<ref>[https://www.matem.unam.mx/~omar/no-wandering-domains.pdf Iteration of Rational Functions by Omar Antolín Camarena]</ref> on the on the [[Riemann sphere]] creates [[Discrete dynamical system|discrete dynamical systems]].
-===Notion of a rational function on an algebraic variety===
-{{Main|Function field of an algebraic variety}}
-Like [[Polynomial ring#The polynomial ring in several variables|polynomials]], rational expressions can also be generalized to ''n'' indeterminates ''X''<sub>1</sub>,..., ''X''<sub>''n''</sub>, by taking the field of fractions of ''F''[''X''<sub>1</sub>,..., ''X''<sub>''n''</sub>], which is denoted by ''F''(''X''<sub>1</sub>,..., ''X''<sub>''n''</sub>).
-An extended version of the abstract idea of rational function is used in algebraic geometry. There the [[function field of an algebraic variety]] ''V'' is formed as the field of fractions of the [[coordinate ring]] of ''V'' (more accurately said, of a Zariski-dense affine open set in ''V''). Its elements ''f'' are considered as regular functions in the sense of algebraic geometry on non-empty open sets ''U'', and also may be seen as morphisms to the [[projective line]].
-==Applications==
-Rational functions are used in [[numerical analysis]] for [[interpolation]] and [[approximation]] of functions, for example the [[Padé approximation]]s introduced by [[Henri Padé]]. Approximations in terms of rational functions are well suited for [[computer algebra system]]s and other numerical [[software]]. Like polynomials, they can be evaluated straightforwardly, and at the same time they express more diverse behavior than polynomials. <!-- Care must be taken, however, since small errors in denominators close to zero can cause large errors in evaluation. -->
-Rational functions are used to approximate or model more complex equations in science and engineering including fields and forces in physics, spectroscopy in analytical chemistry, enzyme kinetics in biochemistry, electronic circuitry, aerodynamics, medicine concentrations in vivo, wave functions for atoms and molecules, optics and photography to improve image resolution, and acoustics and sound{{Citation needed|date=April 2017}}.
-In [[signal processing]], the [[Laplace transform]] (for continuous systems) or the [[z-transform]] (for discrete-time systems) of the [[impulse response]] of commonly-used [[linear time-invariant system]]s (filters) with [[infinite impulse response]] are rational functions over complex numbers.
-==See also==
-* [[Field of fractions]]
-* [[Partial fraction decomposition]]
-* [[Partial fractions in integration]]
-* [[Function field of an algebraic variety]]
-* [[Algebraic fraction]]s{{snd}}a generalization of rational functions that allows taking integer roots
 ==References==
 {{Reflist}}
-*{{springer|id=Rational_function&oldid=17805|title=Rational function}}
+*Martin J. Corless, Art Frazho, Linear Systems and Control, p. 163, CRC Press, 2003 ISBN 0203911377.
-*{{Citation |last1=Press|first1=W.H.|last2=Teukolsky|first2=S.A.|last3=Vetterling|first3=W.T.|last4=Flannery|first4=B.P.|year=2007|title=Numerical Recipes: The Art of Scientific Computing|edition=3rd|publisher=Cambridge University Press| publication-place=New York|isbn=978-0-521-88068-8|chapter=Section 3.4. Rational Function Interpolation and Extrapolation|chapter-url=http://apps.nrbook.com/empanel/index.html?pg=124}}
+*Malcolm W. Pownall, Functions and Graphs: Calculus Preparatory Mathematics, p. 203, Prentice-Hall, 1983 ISBN 0133323048.
+*Glisson, Tildon H., Introduction to Circuit Analysis and Design, Springer, 2011 ISBN ISBN 9048194431.
-==External links==
+*"Rational function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
-* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Rational_functions Dynamic visualization of rational functions with JSXGraph]
+*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
-[[Category:Algebraic varieties]]
-[[Category:Morphisms of schemes]]
-[[Category:Meromorphic functions]]
-[[Category:Rational functions| ]]

Difference between revisions of "Graphs of Rational Functions"

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