The domain of definition of the function `f(x)=sqrt(sin^(-1)(2x)+pi/6)` for real-valued `x` is (a) `[-1/4,1/2]` (b) `[-1/2,1/2]` (c) `(-1/2,1/9)` (d) `[-1/4,1/4]`

Answer

`[-1//4,1//2]`

Explanation

We have `f(x)=sqrt(sin^(-1)(2x)+(pi)/(6))`
We must have `sin^(-1)(2x) +(pi)/(6) ge 0`
`implies sin^(-1)(2x) ge -(pi)/(6) " …(1) " `
But `-(pi)/(2) le sin^(-1)(2x) le (pi)/(2) " (2)" `
From (1) and (2), we have
` -(pi)/(2) le sin^(-1) (2x) le (pi)/(2)`
`implies "sin"(-(pi)/(6)) le 2x le "sin"(pi)/(2)`
`implies -(1)/(2) le 2x le 1`
`implies -(1)/(4) le x le (1)/(2)`
Hence, domain is `[-(1)/(4),(1)/(2)]`