1) y= 2sinx
-2<= 2sinx<=2
Since the range of sinx is between -1 and 1, so doubling it will also double the range
2) y= (sinx) ^2
-1<=(sinx) ^2<=1
1^2 is 1. so squaring the function whose range is between [-1,1] wont change the range. this wouldn't be true if the range were bigger than [-1,1] interval.
3) y= 2^sin(x)
Maximum value of sin(x) is when x = (4n+1)pi/2 and the value is 1. so 2^1 = 2 maximum value
Minimum value of sinx is when x = (4n-1)pi/2 and value is -1. so 2^-1 = 0.5
So the range is 0.5<=2^sinx<=2
4) y= 1/sinx
1/sinx = cosec(x)
and range of cosec(x) = R - (-1,1)
or
(-∞,-1) U (1,∞)