# Find Domain and Range of Arcsine Functions

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]

## More References and Links to Inverse Trigonometric Functions

Inverse Trigonometric Functions
Graph, Domain and Range of Arcsin function
Graph, Domain and Range of Arctan function
Find Domain and Range of Arccosine Functions
Find Domain and Range of Arcsine Functions
Solve Inverse Trigonometric Functions Questions