Consider two functions
`f(x)={([x]",",-2le x le -1),(|x|+1",",-1 lt x le 2):} and g(x)={([x]",",-pi le x lt 0),(sinx",",0le x le pi):}`
where [.] denotes the greatest integer function.
The exhaustive domain of `g(f(x))` is

Options

A. `(sqrt(3))/(4)+(pi)/(6)` sq. units
B. `(sqrt(3))/(2)+(pi)/(6)` sq. units
C. `(sqrt(3))/(4)-(pi)/(6)` sq. units
D. `(sqrt(3))/(2)-(pi)/(6)` sq. units

Answer

A

Explanation

`A=2overset(1//2)underset(0)intsqrt(1-x^(2))dx`
`=2[(x)/(2)sqrt(1-x^(2)):|_(0)^((1)/(2))+(1)/(2) sin^(-1)x:|_(0)^((1)/(2))]`
`= (sqrt(3))/(4)+(p)/(6)` sq. unit.