Objectives: In this tutorial, we define rational functions. We discuss
the domain and range of rational functions. The graph of a rational function
is compared to the graph of the rational function k(x) = c x^{p} for some particluar c and p. The graphs of
f(x) = x^{ n} and g(x) = x^{ m}
are also compared. After working through these materials, the student should be able
 to recognize a rational function algebraically;
 to describe the "longterm behavior" of rational functions;
 to distinguish between x^{n} and x^{m} when
n and m
are different;
Modules:
Definition. A rational function f
is a quotient:
where g and h are polynomials.


 Comments:
 The domain of f consists of all real numbers x such that the denominator h(x)
is not equal to 0.
 It is not easy to give a general description of the range of f. See the first example below.
 Examples:
Discussion: [LiveMath notebook]
 Suppose that g(x) = a_{n}x^{n} + ... + a_{1}x^{1} + a_{0}
and h(x) = b_{m}x^{m} + ... + b_{1}x^{1} + b_{0} with both of a_{n} and b_{m} not equal to zero. If you zoom out sufficiently far, then the graph of the polynomial function f looks just like the function
k(x) = a_{n}/b_{m} x^{n  m}. Use
this LiveMath notebook to experiment to see that this is true.
 You should become familiar with the differences of the graphs of the functions f(x) = x^{ n} and g(x) = x^{ m} for n not equal to m.
Use
this LiveMath notebook to experiment with different values of n and m.