The range of the function f (x) = sin^ - 1 [x^ 2 - (1)/(3)] - cos^...

\left[x^{2} - \dfrac{1}{3}\right] =\left[x^{2} + \dfrac{2}{3} -1\right] = \left[x^{2} + \dfrac{2}{3}\right]$-$1 = $k\left(let\right)$
Now, the function is $sin^{- 1}k - cos^{- 1} \left(k + 1\right)$

For the above function to be defined,
- 1 ≤ k ≤ 1 & - 1 \le \left(k + 1\right)≤ 1$<br />$∴ k =\left\{- 1,0\right\}$<br />$∴$Range$= \left\{sin^{- 1}\left(-1\right)$- cos^{- 1}\left(0\right)$, sin^{- 1}\left(0\right)- cos^{- 1}\left(1\right)\right\}= \left\{- \pi,0\right\}