Graphing arcsin(x) functions
A step by step tutorial on graphing and sketching arcsin functions
where also the domain and range of these functions and other properties are discussed.
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

