Section 4.3
Rational Functions I
447
15.
In
F
(
x
)
=
3
x
(
x

1)
2
x
2

5
x

3
, the denominator,
q
(
x
)
=
2
x
2

5
x

3
=
(2
x
+
1)(
x

3), has zeros
at

1
2
and 3.
Thus, the domain of
F
(
x
) is
x x
≠ 
1
2
,
x
≠
3
.
16.
In
Q
(
x
)
=

x
(1

x
)
3
x
2
+
5
x

2
, the denominator,
q
(
x
)
=
3
x
2
+
5
x

2
=
(3
x

1)(
x
+
2), has zeros
at
1
3
and

2.
Thus, the domain of
Q
(
x
) is
x x
≠ 
2,
x
≠
1
3
.
17.
In
R
(
x
)
=
x
x
3

8
, the denominator,
q
(
x
)
=
x
3

8
=
(
x

2)(
x
2
+
2
x
+
4) , has a zero at 2.
(
x
2
+
2
x
+
4 has no real zeros.)
Thus, the domain of
R
(
x
) is
x x
≠
2
{ }
.
18.
In
R
(
x
)
=
x
x
4

1
, the denominator,
q
(
x
)
=
x
4

1
=
(
x

1)(
x
+
1)(
x
2
+
1), has zeros at –1
and 1.
(
x
2
+
1 has no real zeros.)
Thus, the domain of
R
(
x
) is
x x
≠ 
1,
x
≠
1
{ }
.
19.
In
H
(
x
)
=
3
x
2
+
x
x
2
+
4
, the denominator,
q
(
x
)
=
x
2
+
4 , has no real zeros.
Thus, the domain of
H
(
x
) is
x x
is a real number
{ }
.
20.
In
G
(
x
)
=
x

3
x
4
+
1
, the denominator,
q
(
x
)
=
x
4
+
1, has no real zeros.
Thus, the domain of
G
(
x
) is
x x
is a real number
{ }
.
21.
In
R
(
x
)
=
3(
x
2

x

6)
4(
x
2

9)
, the denominator,
q
(
x
)
=
4(
x
2

9)
=
4(
x

3)(
x
+
3) , has zeros at
3 and –3.
Thus, the domain of
R
(
x
) is
x x
≠ 
3,
x
≠
3
{ }
.
22.
In
F
(
x
)
=

2(
x
2

4)
3(
x
2
+
4
x
+
4)
, the denominator,
q
(
x
)
=
3(
x
2
+
4
x
+
4)
=
3(
x
+
2)
2
, has a zero
at –2.
Thus, the domain of
F
(
x
x x
≠ 
2
{ }
.
23.
(a)
Domain:
x x
≠
2
{ }
;
Range:
y y
≠
1
{ }
(b)
Intercept:
(0, 0)
(c)
Horizontal Asymptote:
y
=
1
(d)
Vertical Asymptote:
x
=
2
(e)
Oblique Asymptote:
none
24.
(a)
Domain:
x x
≠ 
1
{ }
;
Range:
y y
0
{ }
(b)
Intercept:
(0, 2)
(c)
Horizontal Asymptote:
y
=
0
(d)
Vertical Asymptote:
x
= 
1
(e)
Oblique Asymptote:
none