In what follows, arcsin(x) is the inverse function of f(x) = sin(x) for pi/2 ≤ x ≤ pi/2.
The domain of y = arcsin(x) is the range of f(x) = sin(x) for pi/2 ≤ x ≤ pi/2 and is given by the interval [1 , 1]. The range of arcsin(x) is the domain of f which is given by the interval [pi/2 , pi/2]. The graph, domain and range of both f(x) = sin(x) for pi/2 ≤ x ≤ pi/2 and arcsin(x) are shown below.
A table of values of arcsin(x) can made as follows:
x 
1 
0 
1 
y = arcsin(x) 
π/2 
0 
π/2 
Note that there are 3 key points that may be used to graph arcsin(x). These points are: (1,pi/2) , (0,0) and (1,pi/2).
Example 1: Find the domain and range of y = arcsin(x  2) and graph it.
Solution to Example 1:
The graph of y = arcsin(x  2) will be that of arcsin(x) shifted 2 units to the right. The domain is found by stating that 1 ≤ x  2 ≤ 1. Solve the double inequality to find the domain:
1 ≤ x ≤ 3
The 3 key points of arcsin(x) can also be used in this situation as follows:
x  2 
1 
0 
1 
y = arcsin(x2) 
π/2 
0 
π/2 
x 
1 
2 
3 
The value of x is calculated from the value of x  2. For example when x  2 = 1, solve for x to find x = 1 and so on.
The domain is given by the interval [1,3] and the range is given by the interval [pi/2,pi/2]
The three points will now be used to graph y = arcsin(x  2).
Example 2: Find the domain and range of y = 2 arcsin(x + 1) and graph it.
Solution to Example 2:
We use the 3 key points in the table as follows, then find the value 2 arcsin(x + 1) and x.
x + 1 
1 
0 
1 
arcsin(x+1) 
π/2 
0 
π/2 
y = 2 arcsin(x+1) 
π 
0 
π 
x 
2 
1 
0 
domain = [2,0] , range = [ pi , pi]
The graph is that of arcsin(x) shifted one unit to the left and stretched vertically by a factor of 2.
Example 3: Find the domain and range of y =  arcsin(x  1) and graph it.
Solution to Example 3:
We use the 3 key points in the table as follows, then find the value  arcsin(x  1) and x.
x  1 
1 
0 
1 
arcsin(x1) 
π/2 
0 
π/2 
y =  arcsin(x1) 
π/2 
0 
π/2 
x 
0 
1 
2 
domain = [0 , 2] , range = [ pi/2 , pi/2]
The graph is that of arcsin(x) shifted one unit to the right and reflected on the x axis.
More references and links on Graphing Functions
