Rational Function

If , where P(x) and Q(x) are polynomial functions, then f(x) is said to be a rational function.

Objective: We want to know how to graph a rational function.

Preparations: You should know how to find the x, and y- intercepts, and how to solve inequalities.

Example 1: Let Graph f.

First, we set denominators we get x=1 and x=2, these are called the vertical asymtotes of f.

In short, these are vertical lines that the graph of f will get very close to. So let's investigate the following cases:

Case 1: As (this means x approaches to 1 from the right, say x=1.0001), we get (this measns that the outputs will tend to

Case 2: As (this means x approaches to 1 from the left, say x=0.999, note that it does not mean x approaches to -1), we get

Case 3: As say x=2.0001, we get

Case 4: As say x=1.9999, we get

Horizontal Asymtote (h.a.): This is a horizontal line that the graph of f will be very close to when x goes to positive infinity or negative infinity.

To find the h.a., you need to do the following two steps:

• let x be a positive large number (say, x = 10,000) and
• let x be a negative large number (say, x = -10,000). Notice that for the previous problem, when x -> + infinity, f(x) -> 0⁺ , which means that the graph will approach to 0 from above, same is true when x -> -infinity. Thus, y = 0 is a horizontal asymtote for f.

Now, you can use Maple to graph the function. [Maple syntax: plot(2/((x-1)*(x-2)), x = -1..3, y = -50..50, thickness=3); ]

Notice your graph indicates that f(x)>0 if x is in (-infinity, 1) union (2, infinity), and f(x)<0 if x is in (1,2). Do you know why? Can you solve the inequalities f(x)>0 and f(x)<0 by hand?