Section 5.2: Properties of Rational Functions
• Def: A rational function is a function of the form R(x) = p(x)
, where p(x)
q(x)
and q(x) are polynomial functions and q(x) is not the zero polynomial. The
domain of a rational function is the set of all real numbers except those for
which q(x) = 0.
• ex. Find the domain of f (x) =
• The graph of f (x) =
1
x2
• ex. Graph: f (x) = 1 −
−x2 −7x−12
.
2x2 +7x−4
looks like:
1
(x−3)2
1
• Def: A horizontal asymptote of the graph of a function f (x) is a horizontal
line y = L such that as x approaches ∞ or −∞, f (x) approaches L. Usually
we write this as “f (x) → L as x → ±∞” or limx→±∞ f (x) = L.
• Def: A vertical asymptote of the graph of a function f (x) is a vertical line
x = c such that as x approaches c, the value of f (x) approaches ∞ or −∞.
• A horizontal asymptote describes the end behavior of a function f . The graph
of a function f can intersect a horizontal asymptote but it can never intersect
the vertical asymptote.
• ex. Find the vertical and horizontal asymptotes of f (x) =
1
x−2
+ 1.
• Def: A slant or oblique asymptote is an asymptote which is neither vertical
or horizontal. It is a line y = mx + b which f approaches as x approaches ∞
or −∞.
• If a rational function R(x) = p(x)
is in lowest terms (meaning that p(x) and
q(x)
q(x) have no common factors) then R will have a vertical asymptote for each
value of x for which q(x) = 0.
• Fact: Let R(x) =
p(x)
q(x)
=
an xn +...+a0
bm xm +...+b0
be a rational function.
1. If the degree of q(x) is greater than the degree of p(x) (so m > n) then
the line y = 0 is a horizontal asymptote of the graph of R.
2. If the degree of q(x) equals the degree of p(x) (so m = n) then the line
y = abnn is a horizontal asymptote of the graph of R.
3. If the degree of q(x) is one less than the degree of p(x) (so m = n−1) then
use long division to rewrite R(x) as R(x) = mx + b + r(x), where r(x) is
the remainder you get from long division. Then the line y = mx + b is
the oblique asymptote.
4. In all other cases, there are no horizontal or oblique asymptotes.
• ex. Find the vertical, horizontal, and oblique asymptotes, if any, of the
following rational functions:
2
a) G(x) =
−x2 +1
x+5
b) F (x) =
−2x2 +1
2x3 +4x2
3