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

206 Chapter 2 Polynomial and Rational Functions
1. fx
(a)
1
x 1
x
0.5
0.9
0.99
0.999
3. fx
(a)
3x2
x2 1
x f x x
0.5
0.9
0.99
0.999
1
12.79
147.8
1498
4. f x
(a)
4x
x2 1
x f x x
0.5
2.66
0.9 18.95
0.99 199
0.999 1999
1.5
1.1
1.01
1.001
1.5 4.8
f x x
5.4
17.29
152.3
1502
1.1 20.95
1.01 201
1.001 2001
5. fx
Domain: all real numbers except x 0
1
x2 Vertical asymptote:
f x x
2
10
100
1000
2. f x
(a)
5x
x 1
x f x x
0.5 5
0.9 45
0.99 495
0.999 4995
x 0
1.5
1.1
1.01
1.001
1.5 15
1.1 55
Horizontal asymptote: y 0
Degree of Nx < degree of Dx
2
10
100
1000
1.01 505
1.001 5005
f x x
f x x
f x x
5
5
10
100
1000 0.001
10
5
10
100
f x
0.25
0.1
0.01
5 6.25
10
100
f x
5.55
5.05
1000 5.005
1000 3
f x
0.833
0.40
100 0.04
1000 0.004
f x
3.125
3.03
3.0003
(b) The zero of the denominator is x 1, so x 1 is a
vertical asymptote. The degree of the numerator is less
than the degree of the denominator so the x-axis, or
y 0, is a horizontal asymptote.
(c) The domain is all real numbers except x 1.
(b) The zero of the denominator is so is a
vertical asymptote. The degree of the numerator is equal
to the degree of the denominator, so the line y
is a horizontal asymptote.
5
x 1, x 1
1 5
(c) The domain is all real numbers except x 1.
(b) The zeros of the denominator are so both
are vertical asymptotes. Since
the degree of the numerator equals the degree
of the denominator, y is a horizontal
asymptote.
3
x ±1
x 1 and x 1
1 3
(c) The domain is all real numbers except x ±1.
(b) The zeros of the denominator are x ±1 so both
x 1 and x 1 are vertical asymptotes.
Because the degree of the numerator is less than
the degree of the denominator, the x-axis or y 0
is a horizontal asymptote.
(c) The domain is all real numbers except x ±1.
4
6. fx
x 2
Domain: all real numbers except x 2
3
Vertical asymptote:
x 2
Horizontal asymptote: y 0
Degree of Nx < degree of Dx