# Rational functions

**Rational functions** have a polynomial function in the numerator and one in the denominator. The function therefore has the form:

$f(x)=\frac{g(x)}{h(x)}$ $=\frac{a_n\cdot x^n+a_{n-1}\cdot x^{n-1}+...+a_1\cdot x+a_0}{b_m\cdot x^m+b_{m-1}\cdot x^{m-1}+...+b_1\cdot x+b_0}$

$g(x)$ is called **polynomial of the numerator** and $h(x)$ is called **polynomial of the denominator**, since both are polynomials (= function terms of polynomial functions).

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Otherwise it is called

**proper rational function**is a rational function in which the degree of $g(x)$ is less than the degree of $h(x)$.Otherwise it is called

**improper**.### Examples

Examples of rational functions are:

- $f(x)=\frac{x^3}{x-1}$
- $f(x)=\frac{x-2}{x^3+x}$
- $f(x)=\frac{x^4-3x+5}{x^2+5x-4}$