# 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]

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

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

where ${\displaystyle P\,}$ and ${\displaystyle Q\,}$ are polynomials in ${\displaystyle x\,}$ and ${\displaystyle Q\,}$ is not the zero polynomial. The domain of a function|domain of ${\displaystyle f\,}$ is the set of all points ${\displaystyle x\,}$ for which the denominator ${\displaystyle Q(x)\,}$ is not zero.

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

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

which may have a larger domain than ${\displaystyle f(x)}$, and is equal to ${\displaystyle f(x)}$ on the domain of ${\displaystyle f(x).}$ It is a common usage to identify ${\displaystyle f(x)}$ and ${\displaystyle f_{1}(x)}$, that is to extend "by continuity" the domain of ${\displaystyle f(x)}$ to that of ${\displaystyle f_{1}(x).}$ 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 ${\displaystyle {\frac {P(x)}{Q(x)}}}$ is equivalent to ${\displaystyle {\frac {P_{1}(x)}{Q_{1}(x)}}}$.

## Examples[1]

Examples of rational functions
Rational function of degree 3: ${\displaystyle y={\frac {x^{3}-2x}{2(x^{2}-5)}}}$
Rational function of degree 2: ${\displaystyle y={\frac {x^{2}-3x-2}{x^{2}-4}}}$

The rational function ${\displaystyle f(x)={\frac {x^{3}-2x}{2(x^{2}-5)}}}$ is not defined at ${\displaystyle x^{2}=5\Leftrightarrow x=\pm {\sqrt {5}}}$. It is asymptotic to ${\displaystyle {\frac {x}{2}}}$ as x approaches infinity.

The rational function ${\displaystyle f(x)={\frac {x^{2}+2}{x^{2}+1}}}$ is defined for all real numbers, but not for all complex numbers, since if x were a square root of ${\displaystyle -1}$ (i.e. the imaginary unit or its negative), then formal evaluation would lead to division by zero: ${\displaystyle f(i)={\frac {i^{2}+2}{i^{2}+1}}={\frac {-1+2}{-1+1}}={\frac {1}{0}}}$, which is undefined.

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

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

The rational function ${\displaystyle f(x)={\frac {x}{x}}}$ 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 a rational function

(1)Let's sketch the graph of ${\displaystyle y={\frac {1}{x}}}$.
First, we must avoid ${\displaystyle x=0}$ 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:

${\displaystyle x=1}$ ${\displaystyle y=1}$
${\displaystyle x=2}$ ${\displaystyle y={\frac {1}{2}}}$
${\displaystyle x=3}$ ${\displaystyle y={\frac {1}{3}}}$
${\displaystyle x=-3}$ ${\displaystyle y=-{\frac {1}{3}}}$
${\displaystyle x=-2}$ ${\displaystyle y=-{\frac {1}{2}}}$
${\displaystyle x=-1}$ ${\displaystyle y=-1}$

As x get large the function itself gets smaller and smaller. Here is the graph of ${\displaystyle {\frac {1}{x}}}$.

## References

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