Section 4.1

Rational Functions and Graphs
Rational functions = ?????
Recall that a
rational number
is a number that can be written as the ratio
of two integers.
A
rational expression
was the ratio
of two polynomials. For ex.
2
3
5
4
x
x
x

+
+
.
So it makes sense to define a
rational function
as the ratio
of two polynomial functions.
DEFINITION: A rational function is a function
f
that is the quotient of two
polynomials, that is ,
( )
( )
( )
p x
f x
q x
=
,
where p(x) and q(x) are polynomials and q(x)
is not equal to zero.
Ex. The following are some examples of rational functions:
1
( )
f x
x
=
1
( )
1
x
g x
x
+
=

3
2
3
5
( )
4
x
x
h x
x
+
=

An important idea is that the domain of a rational function is all real numbers, except
those that make the denominator zero. So the domains of the functions above are:
f(x)
x
≠
0
g(x)
x
≠
1
and
h(x)
x
≠
2 and x
≠
2
or using interval notation :
f(x)
(
,1)
(1,
)
∞
+∞
U
g(x)
(
,1)
(1,
)
∞
+∞
U
and
h(x)
(
29
,
2)
( 2,2
(2,
)
∞ 

+∞
U
U
HORIZONTAL ASYMPTOTES:
Note the graph of
1
( )
f x
x
=
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 Spring '10
 stean
 Fraction, Limit of a function, Rational function, 0 g, asymptotically.

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