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 xp 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 "long-term behavior" of rational functions;
- to distinguish between x-n and x-m when
n and m
Definition. A rational function f
is a quotient:
where g and h are polynomials.
- 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.
Discussion: [LiveMath notebook]
- Suppose that g(x) = anxn + ... + a1x1 + a0
and h(x) = bmxm + ... + b1x1 + b0 with both of an and bm 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) = an/bm xn - 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.
this LiveMath notebook to experiment with different values of n and m.