Define another rational function with equal zeros in the numerator and denominator and check that the graph is that of a horizontal line.
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.
- g(x) = (x - 1) / (x - 2)
- h(x) = (x + 2) / x
- 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).
- 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).
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?
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.
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).
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.
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.
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