Rational Functions

Rational functions and the properties of their graphs such as domain, vertical and horizontal asymptotes, x and y intercepts are explored using an applet. The investigation of these functions is carried out by changing parameters included in the formula of the function. Each parameter can be changed continuously which allows a better understanding of the properties of the graphs of these functions. Once you finish with the present tutorial, you may want to another tutorial on rational functions to further explore the properties of these functions.

Definition and Domain of Rational Functions

A rational function is defined as the quotient of two polynomial functions.

f(x) = P(x) / Q(x)

Here are some examples of rational functions:

  • g(x) = (x2 + 1) / (x - 1)
  • h(x) = (2x + 1) / (x + 3)

The rational functions to explored in this tutorial are of the form

f(x) = (ax + b)/(cx + d)


where a, b, c and d are parameters that may be changed, using sliders, to understand their effects on the properties of the graphs of rational functions defined above.

Example: Find the domain of each function given below.

  1. g(x) = (x - 1) / (x - 2)
  2. h(x) = (x + 2) / x

Solution

  1. For function g to be defined, the denominator x - 2 must be different from zero or x not equal to 2. Hence the domain of g is given by
    (-infinity , 2)U(2,+infinity).

  2. For function h to be defined, the denominator x must be different from zero or x not equal to 0. Hence the domain of h is given by
    (-infinity , 0)U(0,+infinity).

Interactive Tutorial

Explore rational functions using the link below.

An HTML5 Applet to Explore Rational Functions and Their Asymptotes

There is also another HTML5 Applet to Explore Rational Functions and Their Transforms.

Holes in the Graphs of Rational Functions

What if the zeros of the numerator and the denominator of the rational function are equal?

Example
f(x) = (2x + 2) / (x + 1)
= 2(x + 1) / (x + 1)
= 2 , for x not equal to -1.

The graph of function f is a horizontal line with a hole (function not defined) at x = -1.

  • Define another rational function with equal zeros in the numerator and denominator and check that the graph is that of a horizontal line.

    Vertical Asymptotes of Rational Functions

    Let f(x) = 1/x. f(x) is not defined at x = 0 (division by zero is not allowed). However what is the behavior of the graph "close" to zero?

    In the tables below are values of function f as x approaches zero from the right (x >0) and as x approaches zero from the left (x < 0).

    table of values for f(x) as x approaches zero from right

    We note that as x approaches zero from the right, f(x) takes larger values. Is there a limit to the values of f(x)? No, f(x) increases without bound.

    table of values for f(x) as x approaches zero from left

    We also note that as x approaches zero from the left, f(x) takes smaller values. Is there a limit to the values of f(x)? No, f(x) decreases without bound. The vertical line x = 0 is called the vertical asymptote and it is given by the zero of the denominator.

    Horizontal Asymptotes of Rational Functions

    Let f(x) = 1/x. What is the behavior of the graph of f as |x| becomes very large?

    Tales below show values of f when x becomes very large, and when x becomes very small.

    table of values for f(x) as x takes large values


    table of values for f(x) as x takes small values

    As x takes smaller values or as x takes larger values, f(x) takes values close to zero and the graph approaches the line horizontal line y = 0. This line is called the horizontal asymptote.

    More on topics related to rational functions

    tutorial on rational functions.

    Graphs of rational functions

    tutorial on graphs of rational functions

    self test on graphs of rational functions.

    Solver to Analyze and Graph a Rational Function