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 "long-term 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_nxⁿ + ... + a₁x¹ + a₀ and h(x) = b_mx^m + ... + b₁x¹ + b₀ 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.