Rational Function
If 
, where P(x) and Q(x) are polynomial
functions, then f(x) is said to be a rational funtion.
Objective: We want to know how to graph a rational function.
Example 1: Let 
Graph f.
(1) Vertical Asymptote(s).
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:
(a) As 
(this means x approaches to 1 from the
right, say x=1.0001), we get 
(this measns that the outputs will tend to 
(b) 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 
(c) As 
say x=2.0001, we get 
(d) As 
say x=1.9999, we get 
(2) Horizontal Asymptote:
As 
, say x=105, 
and as 
say x=-105, 
Therefore the horizontal asymtote for f is y=0.
(3) Find the interval(s) where f is increasing or decreasing.
We rewrite 
(note. I mistyped "3", it should be "2") so 
("3" should be "2" here
again) (chain rule).
Thus, 
("3" should be "2" again) Now we need to make a table for
(because 
is not in standard form, so we can't apply the short
cut to solve the inequality).
Thus, f is increasing in 
and decreasing in 
and f has a maximum at 
(note: it should be increasing in (-infinity, 3/2) union (2, infinity).
It decreases in (3/2,2).)
(4) Sketch the graph of f.
Let's use to graph the function. Maple syntax: plot(2/((x-1)*(x-2)), x = -1..3, y = -50..50);