Rational functions

A rational function is a function that can be written as the ratio of two polynomials. In other words, it looks like $f(x) = \frac{P(x)}{Q(x)}$, 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 the denominator zero.

Important properties

• The domain excludes values of $x$ that make the denominator 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 degree of the denominator, the function approaches zero as $x$ becomes very large or very small.

• If the degrees are equal, the function approaches the ratio of the leading coefficients.

• If the numerator's degree is higher than the denominator's, the function grows without bound as $x$ increases or decreases.