Rational Function

Function of the form ${f{{\left({x}\right)}}}=\frac{{{Q}{\left({x}\right)}}}{{{P}{\left({x}\right)}}}=\frac{{{a}_{{0}}{{x}}^{{n}}+{a}_{{1}}{{x}}^{{{n}-{1}}}+\ldots+{a}_{{{n}-{1}}}{x}+{a}_{{n}}}}{{{b}_{{0}}{{x}}^{{m}}+{b}_{{1}}{{x}}^{{{m}-{1}}}+\ldots+{b}_{{{m}-{1}}}{x}+{b}_{{m}}}}$, where ${Q}{\left({x}\right)}$ and ${P}{\left({x}\right)}$ are polynomials is called rational function.

Domain of this function consists of all ${x}$ such that ${Q}{\left({x}\right)}\ne{0}$.

Simple example of the rational function is ${f{{\left({x}\right)}}}=\frac{{{2}{x}+{1}}}{{{{x}}^{{2}}-{x}-{2}}}=\frac{{{2}{x}+{1}}}{{{\left({x}-{2}\right)}{\left({x}+{1}\right)}}}$. Its domain is all ${x}$ except ${x}={2}$ and ${x}=-{1}$.

This function is shown on the figure.