A rational function is an algebraic function expressed as a quotient.
f(x) =
$\frac{p(x)}{q(x)}$, where p(x) and q(x) are polynomials. The domain of f(x) consists of all values of x such that q(x) $\neq$ 0.
The reciprocal function f(x) =
$\frac{1}{x}$ is the simplest rational function with domain x $\neq$ 0. This is also known as the parent function for the family of all rational functions.
Graphs of PolynomialsWhen compared to the graphs of Polynomials, additional behavioral patterns can be observed in the graphs of rational functions, which are discontinuity and asymptotes.
Breaks are discontinuity seen in the graphs of rational functions at values excluded from the domain of the function.
If the function is given in its simplest form, vertical asymptotes occur at the points where the function is not defined. A vertical asymptote is a vertical line which appears to change the course and position of the graph.
If the function given is not in simplest form, that is when p(x) and q(x) have common factors, a point discontinuity is caused at the excluded value. The graph of the function has a break at the point and it is said to have a hole at the point.
The rational functions have horizontal asymptotes at excluded values in the range of the function. The graph of rational function appears to coincide with a horizontal line (which is the horizontal asymptote) as the value of x becomes larger in either or one of positive and negative directions.


The graph of rational function f(x) = $\frac{x+2}{x^{2}x6}$ is shown on the left. The graph is undefined at x = 3 and x = 2. The graph has the vertical asymptote at x = 3 and a hole at x = 2. Note here (x + 2) is a common factor of the numerator and the denominator.
The graph also has the horizontal asymptote at y = 0, the x axis.

The graph of a rational function can also have a slant asymptote when the degree of the numerator polynomial is greater than that of the denominator.