Intermediate level Mathematics/Rational functions

Rational function is "any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials"[1].

Definition[2]Edit

A function   is called a rational function if and only if it can be written in the form

 

where   and   are polynomials in   and   is not the zero polynomial. The domain of a function|domain of   is the set of all points   for which the denominator   is not zero.

However, if   and   have a non constant polynomial greatest common divisor  , then setting   and   produces a rational function

 

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

Examples[1]Edit

Examples of rational functions
Rational function of degree 3:  
Rational function of degree 2:  

The rational function   is not defined at  . It is asymptotic to   as x approaches infinity.

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

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

Every polynomial function   is a rational function with  . A function that cannot be written in this form, such as  , is not a rational function. The adjective "irrational" is not generally used for functions.

The rational function   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 a rational functionEdit

(1)Let's sketch the graph of  .
First, we must avoid   because anything can not be divided by 0. Thus x should not be 0 in the equation. Now we just plug in some values of x. The result is as follows:

   
   
   
   
   
   

As x get large the function itself gets smaller and smaller. Here is the graph of  .

ReferencesEdit

Paul's Online Math Notes"Rational Functions"

  1. a b This section was cited from en:wikipedia Rational functionoldid=591840157
  2. This sentence was cited from en:wikipedia Rational functionoldid=591840157