Lesson 4: Rational Functions

Property

A rational function is a function of the form $R(x) = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomial functions and $q(x) \neq 0$. The domain of a rational function is all real numbers except for those values that make the denominator equal to zero, i.e., where $q(x) = 0$.

Examples

Property

A rational function has the form $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$. An asymptote is a line that the graph of a function approaches but never touches. Rational functions can have vertical asymptotes (where the denominator equals zero) and horizontal asymptotes (determined by the behavior as $x$ approaches infinity).

Examples

Property

To find the range of a rational function $f(x) = \frac{P(x)}{Q(x)}$, set $y = f(x)$ and solve for $x$ in terms of $y$: $x = g(y)$. The range consists of all $y$-values for which this equation has valid solutions (where denominators are non-zero).

Examples