College Algebra 9th txtbk

SECTION 4–4 Rational Functions and Inequalities 307
y
10 f(x) x2 6x 9
x 2 x 2
10
10
x
Z Figure 7
MATCHED PROBLEM 5
Graph f (x)
x 2
x 2 7x 10 .
ZZZ CAUTION ZZZ
The graph of a function cannot cross a vertical asymptote, but the same statement is
not true for horizontal asymptotes. The rational function
f (x) 2x6 x 5 5x 3 4x 2
x 6 1
has the line y 2 as a horizontal asymptote. The graph of f in Figure 8 clearly shows
that the graph of a function can cross a horizontal asymptote. The definition of a horizontal
asymptote requires f (x) to approach b as x increases or decreases without bound,
but it does not preclude the possibility that f (x) b for one or more values of x.
y
4
2x 6 x 5 5x 3 4x 2
f(x)
x 6 1
y 2 is a horizontal asymptote
5
5
x
Z Figure 8 Multiple intersections of a graph and a horizontal asymptote.
EXAMPLE 6 Graphing a Rational Function
Graph f (x) x2 3x 4
.
x 2
SOLUTION
x intercepts: x 1 and x 4
y intercept: f (0) 2
Domain: (, 2) (2, )
Points of discontinuity: x 2
Vertical asymptote: x 2
No horizontal asymptote
f (x) x2 3x 4
x 2
(x 1)(x 4)
x 2