Definition:Rational Function

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Definition

Let $F$ be a field.

Let $P: F \to F$ and $Q: F \to F$ be polynomial functions on $F$.

Let $S$ be the set $F$ from which all the roots of $Q$ have been removed.

That is:

$S = F \setminus \set {x \in F: \map Q x = 0}$


Then the equation $y = \dfrac {\map P x} {\map Q x}$ defines a mapping from $S$ to $F$.


Such a mapping is called a rational function.


The concept is usually encountered where the polynomial functions $P$ and $Q$ are either real or complex:


Real Domain

Let $P: \R \to \R$ and $Q: \R \to \R$ be polynomial functions on the set of real numbers.

Let $S$ be the set $\R$ from which all the roots of $Q$ have been removed.

That is:

$S = \R \setminus \set {x \in \R: \map Q x = 0}$.


Then the equation $y = \dfrac {\map P x} {\map Q x}$ defines a function from $S$ to $\R$.


Such a function is known as a rational function.


Complex Domain

Let $P: \C \to \C$ and $Q: \C \to \C$ be polynomial functions on the set of complex numbers.

Let $S$ be the set $\C$ from which all the roots of $Q$ have been removed.

That is:

$S = \C \setminus \set {z \in \C: \map Q z = 0}$


Then the equation $y = \dfrac {\map P z} {\map Q z}$ defines a function from $S$ to $\C$.


Such a function is a rational (complex) function.


Examples

Arbitrary Example $1$

The function defined as:

$\map f x = \dfrac {x^3 + 2 x + 3} {x + 6}$

is a rational function which is not defined at $x = 6$.


Arbitrary Example $2$

The function defined as:

$\map f x = \dfrac {2 x^2 + 3 x + 4} {x^3 + 2}$

is a rational function which is not defined at $x = -\sqrt [3] 2$.


Also known as

A rational function is also known as:

a rational mapping
a rational map
a rational expression.


Also see

  • Results about rational functions can be found here.


Sources

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