C H A P T E R 2 Polynomial and Rational Functions

129. 3 6i 8 3i 3 6i 8 3i 11 9i
131. 6 2i1 7i 6 42i 2i 14i 2 20 40i
Section 2.6 Rational Functions
Section 2.6 Rational Functions 205
■ You should know the following basic facts about rational functions.
(a) A function of the form where and are polynomials, is called
a rational function.
(b) The domain of a rational function is the set of all real numbers except those which make the denominator zero.
(c) If is in reduced form, and a is a value such that then the line is a
vertical asymptote of the graph of f.
(d) The line is a horizontal asymptote of the graph of f if or
(e) Let fx where Nx and Dx have no common factors.
1. If n < m, then the x-axis y 0 is a horizontal asymptote.
Nx
Dx anxn an1xn1 . . . a1x a0 bmxm bm1xm1 . . . f x NxDx, Dx 0, Nx Dx
f x NxDx
Da 0,
x a
fx→ or fx→ as x→a.
y b
fx → b as x → x → .
b1x b0 2. If then y is a horizontal asymptote.
an n m,
b m
3. If n > m, then there are no horizontal asymptotes.
Vocabulary Check
130. 12 5i 16i 12 11i
132. 9 5i9 5i 81 25i 2 81 25 106
133. gx fx 2 134. gx f x 2 135. gx 2fx
136.
4
3
2
1
−1
−2
y
(0, 0)
(2, 2)
(4, 2)
(6, 4)
2 3 4 5 6
Horizontal shift two units
to the right
gx f x 137.
(−4, 4)
(−2, 2)
−4 −3 −2 −1
−1
−2
Reflection in the y-axis
4
3
1
y
(0, 2)
1
x
(2, 0)
x
2
3
2
(0, 0)
−2 −1 1 2 3 4
−2
(−2, −2)
−3
y
Vertical shift two units downward
1. rational functions 2. vertical asymptote 3. horizontal asymptote 4. slant asymptote
(2, 0)
(4, 2)
gx f2x 138.
(−1, 0)
4
3
(0, 2)
y
Horizontal shrink each x-value is
multiplied by 1
2
(1, 2)
(2, 4)
−2 −1
1 2
x
x
10
8
6
(0, 4)
(−2, 0)
−2
y
(2, 4)
(4, 8)
2 4 6 8
Vertical stretch (each y-value is
multiplied by 2)
gx f 1
2 x
10
8
6
4
(0, 2)
(−4, 0)
−2
y
(4, 2)
(8, 4)
2 4 6 8
Horizontal stretch each
x-value
is multiplied by 2
x
x