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

208 Chapter 2 Polynomial and Rational Functions
19.
23.
25.
f x 1 3
x 3
1 3
0
x 3
x 3 3
f x
Domain: all real numbers x except x 1 and x 3
x2 1
x2 x 1x 1 x 1
, x 1 24.
2x 3 x 1x 3 x 3
Horizontal asymptote:
y 1
Degree of Nx degree of Dx
Vertical asymptote: x 3
Since
x 1 is a common factor of Nx and Dx, x 1
is not a vertical asymptote of f x.
f x x2 3x 4
2x 2 x 1
1 3
x 3
x 6 is a zero of f x.
x 1x 4
2x 1x 1
x 4
, x 1
2x 1
Domain: all real numbers except and
Horizontal asymptote:
Vertical asymptote: x Since
x 1 is a common
factor of Nx and Dx, x 1 is not a vertical asymptote
of f x.
1
y
Degree of Nx degree of Dx
2
1
x x 1
2
1
x
2
20.
26.
gx x3 8
x 2 1
x3 8
x2 0
1
x 3 8 0
x 3 8
x 2 is a real zero of gx.
21.
x 4
f x
x
Domain: all real numbers x except x ±4
Horizontal asymptote: y 0
Degree of Nx < degree of Dx
Vertical asymptote: x 4 Since
x 4 is a common
factor of Nx and Dx, x 4 is not a vertical asymptote
of f x.
2 1
, x 4
16 x 4
22.
x 3
f x
x
Domain: all real numbers x except x ±3
The degree of the numerator is less than the degree of the
denominator, so the graph has the line y 0 as a
horizontal asymptote.
Vertical asymptote: x 3 Since
x 3 is a common
factor of Nx and Dx, x 3 is not a vertical asymptote
of f x.
2 9
x 3 1
, x 3
x 3x 3 x 3
f x x2 4
x 2 3x 2
x 2x 2
x 2x 1
Domain: all real numbers x except x 1 and x 2
Horizontal asymptote:
x 2
, x 2
x 1
y 1
Degree of Nx degree of Dx
Vertical asymptote: x 1 Since
x 2 is a common
factor of Nx and Dx, x 2 is not a vertical asymptote
of f x.
f x 6x2 11x 3
6x 2 7x 3
x 2
2x 33x 1
2x 33x 1
Domain: all real numbers x except x or
Horizontal asymptote: y 1
3
2
Degree of Nx degree of Dx
3x 1 3
, x
3x 1 2
x 1
3
Vertical asymptote: Since is a common
factor of and x is not a vertical asymptote
of f x.
3
x 2x 3
Nx Dx, 2
1
3