Section 4.3
1) √49
This is asking what positive
number times itself is 49.
3) −√81
This is asking me to find the
square root of 81, then multiply by (-1)
Solution: 7
Think of the problem like this
= −1 ∙ √81
= −1 ∙ 9
Solution: - 9
5) 5√9 = 5*3
Solution: 15
3
3
9) √−8
This is asking what number times
itself 3 times is equal to -8.
11) − √−8
The answer is -2 because: (-2)(-2)(-2) = -8
−1 ∙ √−8 = -1 * (-2) = 2
Think of this problem as
3
Solution: -2
Solution: 2
4
13) √1
This is asking what number times
itself 4 times equals one. The answer is 1
because: (1)(1)(1)(1) = 1
9
15) √16
To find the square root of a fraction,
just find the square root of the numerator
and denominator separately.
Solution: 1
Solution:
𝟑
𝟒
3
1
17) √8
To find the cubed root of a fraction,
just find the cubed root of the numerator
and denominator separately.
4
19) √16
This is asking what positive number
times itself 4 times is 16.
The answer is 2.
Since (2)(2)(2)(2) = 16
Solution:
𝟏
Solution: 2
𝟐
4
4
21) 3 √16 = 3*2
23) − √16
Solution: 6
Think of this as
4
−1 ∙ √16
Or -1*2
Solution: -2
27) 4√25
This is just 4*5
(multiply 4 by the square root of 25)
Solution: 20
29)
3
2√27
This is just 2*3
(multiply 2 by the cubed root of 27)
Solution: 6
31 – 42 are calculator only problems
3
47)
√𝑎2
49) √𝑦 3
Since the index 3, is an odd number,
I do not need absolute values with my answer.
Since the index is 2, is an even number,
I need absolute values with my answer.
Solution: y
Solution: |𝒂|
4
5
51) √𝑎4
53) √𝑦 5
Since the index is 4, is an even number,
I need absolute values with my answer.
Since the index 5, is an odd number,
I do not need absolute values with my answer.
Solution: |𝒂|
Solution: y
55)
√25𝑥 2
57) √𝑥 6
Divide the exponent of 6
by the index of 2. x6/2
Find the square root of 25 which is 5,
and divide the exponent of the x by
the index of 2.
Solution: x3
Solution: 5x
59) √𝑧 8
Divide the exponent 8
by the index of 2. z8/2
Solution: z4
4
67) √𝑦 20
Divide the exponent 20
by the index 4.
= y20/4
Solution: y5
64
3
8𝑥 9 𝑦 12
69) √𝑦 8
71) √
Find the square root of 64 which is 8,
leave it in the numerator.
Divide the exponent of the y by 2,
and leave the y in the denominator
Find the cubed root of 8 which is 2.
Divide each of the exponents of
the variables by 3.
𝟖
Solution:
𝑧 15
Solution:
𝟐𝒙𝟑 𝒚𝟒
𝒛𝟓
𝒚𝟒
9
73) √16𝑥 2
Find the square root of 9 and 16. Divide the exponent of the x by two to get the answer.
3
= 4𝑥
𝟑
Solution:𝟒𝒙
3
125𝑥 9
75) √ 64𝑧 6
Find the cubed root of 125 and 64. Divide the exponent of the variables by three to get the
answer.
Solution:
𝟓𝒙𝟑
𝟒𝒛𝟐
79) Use a calculator to complete the table, round to two decimal places when needed. Sketch a
graph of the function and find the domain and range of the function in interval notation.
Let ℎ(𝑥) = √𝑥
x
4
3
2
1
0
-1
-2
h(x)
√4 = 2
√3 = 1.73
√2 = 1.41
√1 = 1
√0 = 0
√−1 𝑛𝑜𝑡 𝑟𝑒𝑎𝑙
√−2 𝑛𝑜𝑡 𝑟𝑒𝑎𝑙
point
(4,2)
(3,1.73)
(2, 1.41)
(1,1)
(0,0)
No point
No point
Domain: The x-coordinate of the far right point is 0. The graph extends to the far right end of
the x-axis so the domain is: [0, ∞)
Range: The y-coordinate of the bottom point is 0. The graph extends to the top of the y-axis, so
the range is also [0, ∞)
81) Use a calculator to complete the table, round to two decimal places when needed. Sketch a
graph of the function and find the domain and range of the function in interval notation.
3
Let ℎ(𝑥) = √𝑥
x
h(x)
3
4
√4 = 1.59
3
3
√3 = 1.44
3
2
√2 = 1.26
3
1
√1 = 1
3
.5
√. 5 = .79
3
0
√0 = 0
3
-.5
√−.5 = −.79
3
-1
√−1 = −1
3
-2
√−2 = −1.26
point
(4,1.59)
(3,1.44)
(2,1.26)
(1,1)
(.5, .79)
(0,0)
(.5, -.79)
(-1,-1)
(-2,-1.26)
Domain: The graph extends to the far left edge of the x-axis and to the far right edge of the xaxis. The domain is (−∞, ∞)
Range: The graph extends to the bottom of the y-axis and to the top of the y-axis. The range is
(−∞, ∞)
83) a) 𝑓(𝑥) = √𝑥 − 4
First I will find the domain.
𝑥 − 4 ≥ 0 (add 4 to each side to get the domain)
Domain 𝑥 ≥ 4 which also can be written as [4, ∞)
This tells me I only need numbers that are 4 or larger in my table.
x
4
5
6
7
8
h(x)
√4 − 4 = 0
√5 − 4 = 1
√6 − 4
= 1.41
√7 − 4
= 1.73
√8 − 4 = 2
point
(4,0)
(5,1)
(6, 1.41)
(7,1.73)
(8,2)
I know the domain is [4, ∞)
I can find the range from the graph. The y-coordinate of the bottom point is y=0. The graph
extends to the top of the y-axis. The range is [0, ∞)
3
83 b) 𝑓(𝑥) = √𝑥 − 4
I will build a table with 4 in the middle, and put a few numbers larger and smaller than 4 in the
x-column. The number that I use in the start of my domain table of a square root will be in the
middle of my tables for my cube root graphs.
x
6
5
4
3
2
h(x)
3
√6 − 4 = 1.26
3
√5 − 4 = 1
3
√4 − 4 = 0
3
√3 − 4 = −1
3
√2 − 4 = −1.26
point
(6,1.26)
(5,1)
(4,0)
(3,-1)
(2,-1.26)
Domain: The graph extends to the far left edge of the x-axis and to the far right edge of the xaxis. The domain is (−∞, ∞)
Range: The graph extends to the bottom of the y-axis and to the top of the y-axis. The range is
(−∞, ∞)
85) a) 𝑓(𝑥) = √𝑥 + 4
First I will find the domain.
𝑥 + 4 ≥ 0 (subtract 4 from each side to get the domain)
Domain 𝑥 ≥ −4 which also can be written as [−4, ∞)
This tells me I only need numbers that are -4 or larger in my table.
x
-4
-3
-2
-1
0
h(x)
√−4 + 4 = 0
√−3 + 4 = 1
√−2 + 4 = 1.41
√−1 + 4 = 1.73
√0 + 4 = 2
point
(-4,0)
(-3,1)
(-2, 1.41)
(-1,1.73)
(0,2)
I know the domain is [−4, ∞)
I can find the range from the graph. The y-coordinate of the bottom point is y=0. The graph
extends to the top of the y-axis. The range is [0, ∞)
3
85 b) 𝑓(𝑥) = √𝑥 + 4
I will build a table with -4 in the middle, and put a few numbers larger and smaller than -4 in the
x-column. The number that I use in the start of my domain table of a square root will be in the
middle of my tables for my cube root graphs.
x
-2
-3
-4
-5
-6
h(x)
3
√−2 + 4 = 1.26
3
√−3 + 4 = 1
3
√−4 + 4 = 0
3
√−5 + 4 = −1
3
√−6 + 4 = −1.26
point
(-2,1.26)
(-3,1)
(-4,0)
(-5,-1)
(-6,-1.26)
Domain: The graph extends to the far left edge of the x-axis and to the far right edge of the xaxis. The domain is (−∞, ∞)
Range: The graph extends to the bottom of the y-axis and to the top of the y-axis. The range is
(−∞, ∞)
87) a) 𝑓(𝑥) = √𝑥 − 4 + 3
First I will find the domain.
𝑥 − 4 ≥ 0 (add 4 to each side to get the domain)
Domain 𝑥 ≥ 4 which also can be written as [4, ∞)
This tells me I only need numbers that are 4 or larger in my table.
x
4
5
6
7
8
h(x)
√4 − 4 + 3 = 3
√5 − 4 + 3 = 4
√6 − 4 + 3 = 4.41
√7 − 4 + 3 = 4.73
√8 − 4 + 3 = 5
point
(4,3)
(5,4)
(6, 4.41)
(7,4.73)
(8,5)
I know the domain is [4, ∞)
I can find the range from the graph. The y-coordinate of the bottom point is y=3. The graph
extends to the top of the y-axis. The range is [3, ∞)
3
87) b) 𝑓(𝑥) = √𝑥 − 4 + 1
I will build a table with 4 in the middle, and put a few numbers larger and smaller than 4 in the
x-column. The number that I use in the start of my domain table of a square root will be in the
middle of my tables for my cube root graphs.
x
6
5
4
3
2
h(x)
3
√6 − 4 + 3 = 4.26
3
√5 − 4 + 3 = 4
3
√4 − 4 + 3 = 3
3
√3 − 4 + 3 = 2
3
√2 − 4 + 3 = 1.74
point
(6,4.26)
(5,1)
(4,0)
(3,-1)
(2,1.74)
Domain: The graph extends to the far left edge of the x-axis and to the far right edge of the xaxis. The domain is (−∞, ∞)
Range: The graph extends to the bottom of the y-axis and to the top of the y-axis. The range is
(−∞, ∞)
89) a) 𝑓(𝑥) = √𝑥 + 4 + 2
First I will find the domain.
𝑥 + 4 ≥ 0 (subtract 4 from each side to get the domain)
Domain 𝑥 ≥ −4 which also can be written as [−4, ∞)
This tells me I only need numbers that are -4 or larger in my table.
x
-4
-3
-2
-1
0
h(x)
√−4 + 4 + 2 = 2
√−3 + 4 + 2 = 3
√−2 + 4 + 2 = 3.41
√−1 + 4 + 2 = 3.73
√0 + 4 + 2 = 4
point
(-4,2)
(-3,3)
(-2, 3.41)
(-1,3.73)
(0,4)
I know the domain is [−4, ∞)
I can find the range from the graph. The y-coordinate of the bottom point is y=2. The graph
extends to the top of the y-axis. The range is [2, ∞)
3
89 b) 𝑓(𝑥) = √𝑥 + 4 + 3
I will build a table with -4 in the middle, and put a few numbers larger and smaller than -4 in the
x-column. The number that I use in the start of my domain table of a square root will be in the
middle of my tables for my cube root graphs.
x
-2
-3
-4
-5
-6
h(x)
3
√−2 + 4 + 3 = 4.26
3
√−3 + 4 + 3 = 4
3
√−4 + 4 + 3 = 3
3
√−5 + 4 + 3 = 2
3
√−6 + 4 + 3 = 1.74
point
(-2,4.26)
(-3,4)
(-4,3)
(-5,2)
(-6,1.74)
Domain: The graph extends to the far left edge of the x-axis and to the far right edge of the xaxis. The domain is (−∞, ∞)
Range: The graph extends to the bottom of the y-axis and to the top of the y-axis. The range is
(−∞, ∞)
91) a) 𝑓(𝑥) = √2𝑥 − 6 + 1
First I will find the domain.
2𝑥 − 6 ≥ 0 (add 6 to each side, then divide by two to get the domain)
Domain 𝑥 ≥ 3 which also can be written as [3, ∞)
This tells me I only need numbers that are 3 or larger in my table.
x
3
4
5
6
7
h(x)
√2 ∗ 3 − 6 + 1 = 1
√2 ∗ 4 − 6 + 1 = 2.41
√2 ∗ 5 − 6 + 1 = 3
√2 ∗ 6 − 6 + 1 = 2.64
√2 ∗ 7 − 6 + 1 = 3.83
point
(3,1)
(4,2.41)
(5,3)
(6,2.64)
(7,3.83)
I know the domain is [3, ∞)
I can find the range from the graph. The y-coordinate of the bottom point is y=1. The graph
extends to the top of the y-axis. The range is [1, ∞)
3
91) b) 𝑓(𝑥) = √2𝑥 − 6 + 1
I will build a table with 3 in the middle, and put a few numbers larger and smaller than 3 in the
x-column. The number that I use in the start of my domain table of a square root will be in the
middle of my tables for my cube root graphs.
x
5
4
3
2
1
h(x)
3
√2 ∗ 5 − 6 + 1 = 2.59
3
√2 ∗ 4 − 6 + 1 = 2.26
3
√2 ∗ 3 − 6 + 1 = 1
3
√2 ∗ 2 − 6 + 1 = −0.26
3
√2 ∗ 1 − 6+= −0.59
point
(5,2.59)
(4,2.26)
(3,1)
(2,-0.26)
(1, -0.59)
Domain: The graph extends to the far left edge of the x-axis and to the far right edge of the xaxis. The domain is (−∞, ∞)
Range: The graph extends to the bottom of the y-axis and to the top of the y-axis. The range is
(−∞, ∞)