Rational functions (other)

A rational function is a function that can be written as the ratio of two polynomials. Beyond basic properties, we study how rational functions behave: their graphs, asymptotes, holes, and how to simplify or decompose them.

Important properties

• A rational function has the form $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

• Vertical asymptotes occur at values of $x$ where $Q(x) = 0$ (unless the same factor cancels in $P(x)$).

• Holes in the graph occur where a factor cancels from both $P(x)$ and $Q(x)$.

• Horizontal or oblique asymptotes depend on the degrees of $P(x)$ and $Q(x)$.

• Rational functions can often be simplified by factoring and cancelling common factors.