Rational functions

A rational function is a function that can be written as the ratio of two polynomials. That is, a function $f(x)$ is rational if it can be written as $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$. The domain of a rational function is all real numbers except those that make $Q(x) = 0$.

Important properties

• The domain excludes values where the denominator is zero.

• Rational functions can have vertical asymptotes at points where the denominator is zero (and the numerator is not zero there).

• If the degree of the numerator is less than the denominator, the function approaches zero as $x$ becomes very large or very small.

• If the degree of the numerator is equal to the denominator, the function approaches the ratio of the leading coefficients as $x$ becomes very large or very small.