Rational functions

In mathematics, a 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. The coefficients of the polynomials need not be rational numbers, they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero and the codomain is L.

The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K.

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

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

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