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

232 Chapter 2 Polynomial and Rational Functions
52.
53.
55.
57.
y
y 2x2
x 2 4
−6 6
6
−2
(a) y ≥ 1 when x ≤ 2 or x ≥ 2.
This can also be expressed as x ≥ 2.
(b) y ≤ 2 for all real numbers x.
This can also be expressed as < x < .
Critical numbers:
Test intervals:
Test: Is
2x 2
x 1
14
−15 15
−6
4 x 2 ≥ 0 56.
2 x2 x ≥ 0
4 x 2 ≥ 0?
x ±2
, 2, 2, 2, 2,
By testing an x-value in each test interval in the inequality,
we see that the domain set is: 2, 2
x 2 7x 12 ≥ 0 58.
x 3x 4 ≥ 0
Critical numbers:
Test intervals:
x 3, x 4
Test: Is x 3x 4 ≥ 0?
By testing an x-value in each test interval in the inequality,
we see that the domain set is: , 3 4,
(a)
, 3, 3, 4, 4,
y ≤ 0 (b) y ≥ 8
2x 2
≤ 0
x 1
y ≤ 0 when 1 < x ≤ 2.
54. y
(a)
5x
x2 4
(b)
5x x2 4
x 2 4
x 4x 1
x2 ≥ 0
4
y ≥ 1 when 1 ≤ x ≤ 4.
y ≤ 0
y ≥ 1
5x
x2 ≥ 1
4
≥ 0
5x
x 2 4 ≤ 0 −6 6
y ≤ 0 when < x ≤ 0.
x 2 4 ≥ 0
x 2x 2 ≥ 0
Critical numbers:
x 2, x 2
Test intervals: , 2 ⇒ x 2x 2 > 0
2, 2 ⇒ x 2x 2 < 0
Domain: , 2 2,
144 9x 2 ≥ 0
94 x4 x ≥ 0
Critical numbers:
4
−4
2, ⇒ x 2x 2 > 0
Test intervals: , 4 ⇒ 94 x4 x < 0
Domain: 4, 4
2x 2
≥ 8
x 1
2x 2 8x 1
≥ 0
x 1
6x 12
≥ 0
x 1
6x 2
≥ 0
x 1
y ≥ 8 when 2 ≤ x < 1.
x 4, x 4
4, 4 ⇒ 94 x4 x > 0
4, ⇒ 94 x4 x < 0