# Find Domain and Range of Arcsine Functions

Questions on how to find domain and range of arcsine functions.
## Theorem1.y = arcsin x is equivalent to sin y = xwith -1 ≤ x ≤ 1 and - pi / 2 ≤ y ≤ pi / 2
## Question 1Find the domain and range of y = arcsin(x - 1)
## Question 2Find the domain and range of y = - arcsin(x + 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 3Find the domain and range of y = -2 arcsin(3 x - 1)
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 4Find the domain and range of y = 4 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

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