Rational Functions

A rational function is a function that is a fraction of the form f(x) = P(x)/Q(x) , where p(x) and q(x) are polynomials and q(x) does not equal zero e.g. f(x) = 1/(x+2) , g(x) = 1/(x²-x-2) are examples of rational functions where f and g are the names of the functions and x is the independent variable.

The domain of a rational function consists of all real numbers x except those for which the denominator is zero. When graphing a rational function, we must pay special attention to the behavior of the graph near those x-values. Let us find domain of f(x) = 1/(x+3)

• The function has independent variable (x) in the denominator 
• Set the denominator equal to zero

           x + 3 = 0
           x = -3

• The domain of function f is all real numbers except x = -3 

For a rational function an asymptote is a line that the curve approaches but does not cross. The equations of the vertical asymptotes can be found by finding the zeros of q(x). Completely ignore the numerator when looking for vertical asymptotes, only the denominator matters e.g. for above function f(x) the vertical asymptote is the line x = -3.

Enter different rational functions in the input box of the applet below and try to find the domain , range and asymptote. This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com