Editing Rational function (section)

==Examples==
{{multiple image
 | header = Examples of rational functions
 | align = right
 | direction = vertical
 | width = 300
 | image1 = RationalDegree3.svg
 | alt1 = Rational function of degree 3
 | caption1 = Rational function of degree 3, with a graph of [[degree of an algebraic variety|degree]]&nbsp;3: <math>y = \frac{x^3-2x}{2(x^2-5)}</math>
 | image2 = RationalDegree2byXedi.svg
 | alt2 = Rational function of degree 2
 | caption2 = Rational function of degree 2, with a graph of [[degree of an algebraic variety|degree]]&nbsp;3: <math>y = \frac{x^2-3x-2}{x^2-4}</math>
}}

The rational function

:<math>f(x) = \frac{x^3-2x}{2(x^2-5)}</math>

is not defined at

:<math>x^2=5 \Leftrightarrow x=\pm \sqrt{5}.</math>

It is asymptotic to <math>\tfrac{x}{2}</math> as <math>x\to \infty.</math>

The rational function

:<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:

:<math>f(i) = \frac{i^2 + 2}{i^2 + 1} = \frac{-1 + 2}{-1 + 1} = \frac{1}{0},</math>

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.

Every [[Laurent polynomial]] can be written as a rational function while the converse is not necessarily true, i.e., the ring of Laurent polynomials is a [[subring]] of the rational 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.