Math Real functions Rational functions

# 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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### Info

A 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:

• $f(x)=\frac{x^3}{x-1}$

• $f(x)=\frac{x-2}{x^3+x}$

• $f(x)=\frac{x^4-3x+5}{x^2+5x-4}$