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Questions on how to find domain and range of arcsine functions.
Theorem
1.
y = arcsin x is equivalent to sin y = x
with -1 ≤ x ≤ 1 and - pi / 2 ≤ y ≤ pi / 2
Question 1
Find the domain and range of y = arcsin(x - 1)
Solution to question 1
1. Domain: To find the domain of the above function, we need to impose a condition on the argument (x - 1) according to the domain of arcsin(x) which is -1 ≤ x ≤ 1 . Hence
-1 ≤ (x - 1) ≤ 1
solve to obtain domain as: 0 ≤ x ≤ 2
which as expected means that the graph of y = arcsin(x - 1) is that of y = arcsin(x) shifted one unit to the right.
2. Range: A shift to the right does not affect the range. Hence the range of y = arcsin(x - 1) is the same as the range of arcsin(x) which is - pi / 2 ≤ y ≤ pi / 2
Question 2
Find the domain and range of y = - arcsin(x + 2)
Solution to question 2
1. Domain: To find the domain of the above function, we need to impose a condition on the argument (x + 2) according to the domain of arcsin(x) which is -1 ≤ x ≤ 1 . Hence
-1 ≤ (x + 2) ≤ 1
solve to obatain domain as: - 3 ≤ x ≤ - 1
which as expected means that the graph of y = arcsin(x + 2) is that of y = arcsin(x) shifted two units to the left.
2. Range: The range of arcsin(x + 2) is the same as the range of arcsin(x) which is - pi / 2 ≤ y ≤ pi / 2. Hence we can write
- pi / 2 ≤ arcsin(x + 2)
≤ pi / 2
We now multiply all terms of the above inequality by - 1 and invert the inequality symbols
pi / 2 ≥ - arcsin(x + 2) ≥ - pi / 2
Which is equivalent to
- pi / 2 ≤ - arcsin(x + 2)≤ pi / 2
which gives the range of y = - arcsin(x + 2) as the interval [- pi / 2 , pi / 2]
Question 3
Find the domain and range of y = -2 arcsin(3 x - 1)
Solution to question 3
1. Domain: To find the domain, we need to impose the following condition
-1 ≤ (3 x - 1) ≤ 1
solve to obtain domain as: 0 ≤ x ≤ 2 / 3
2. Range: The range of arcsin(3x - 1) is the same as the range of arcsin(x) which is - pi / 2 ≤ y ≤ pi / 2. Hence we can write
- pi / 2 ≤ arcsin(3x - 1) ≤ pi / 2
We now multiply all terms of the above inequality by - 2 and invert the inequality symbols
pi ≥ - 2 arcsin(3x - 1) ≥ - pi
which gives the range of y = - 2 arcsin(3x - 1) as the interval [- pi , pi]
Question 4
Find the domain and range of y = 4 arcsin( -2(x - 1) ) - pi/2
Solution to question 4
1. Domain: To find the domain, we need to impose the following condition
-1 ≤ -2(x - 1) ≤ 1
solve to obtain domain as: 1 / 2 ≤ x ≤ 3 / 2
2. Range: The range of arcsin(-2(x - 1)) is the same as the range of arcsin(x) which is - pi / 2 ≤ y ≤ pi / 2. Hence we can write
- pi / 2 ≤ arcsin(-2(x - 1)) ≤ pi / 2
We now multiply all terms of the above inequality by 4
-2 pi ≥ 4 arcsin(-2(x - 1)) ≥ 2 pi
We now subtract - pi/2 from all terms of the above inequality.
- 5 pi / 2 ≥ 4 arcsin(-2(x - 1)) ≥ 3 pi / 2
which gives the range of y = 4 arcsin(-2(x - 1)) - pi / 2 as the interval [- 5 pi / 2 , 3 pi / 2]
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