Domain (D) and range (R) of `f(x)=sin^(-1)(cos^(-1)[x]),` where [.] denotes the greatest integer function, is `D-=x in [1,2],R in {0}` D`-=x in 90 ,1],R-={-1,0,1}` `-=x in [-1,1],R-={0,sin^(-1)(pi/2),sin^(-1)(pi)}` `-=x in [-1,1],R-={-pi/2,0,pi/2}`

Options

A. `[1, 2) and {0}`
B. `[0, 1] and {-1, 0, 1}`
C. `[-1, 1] and {0, sin ^(-1)((pi)/(2)),sin^(-1) (pi)}`
D. `[-1 1] and {- (pi)/(2), 0, (pi)/(2)}`

Answer

A

Explanation

Let `f(x) = cos^(-1)[x] and g(x) = sin^(-1) x`. Then, `phi(x) = gof(x)`. Clearly
`D(f) = [-1,1), R(f) = [Cos^(-1)(-1), cos^(-1) (0), cos^(-1)(1)]= {pi,(pi)/(2),0}`
`D(g) = [-1,1] and R(g) =[-(pi)/(2),(pi)/(2)]`
`therefore" "D(phi) = D(gof) = {x : x in D(f) and f(x) in D(g)}`
`={x: x in [-1,2) and f(x)in [-1,1]}`
`{x:x in {-1,2) and cos^(-1) [x] in [-1,1]}`
`=[1,2)`
`R(phi) ={ phi (x) : x in |1,2)}= { sin^(-1) (cos^(-1)1)}={sin^(-1)(0)} = {0}`