Question

The range of the function $f\left(x\right)=sin^{- 1}\left[x^{2} - \frac{1}{3}\right]-cos^{- 1}\left[x^{2} + \frac{2}{3}\right]$ is (where, $\left[x\right]$ represents the greatest integer value of $x$ )

• A. $\left[- \pi , 0\right]$
• B. $\left\{- \pi , 0\right\}$
• C. $\left\{0 , \pi \right\}$
• D. $\left\{0 , \pi , - \pi \right\}$

Answer 1

Answer: (B)

$\left[x^{2} - \frac{1}{3}\right]=\left[x^{2} + \frac{2}{3} - 1\right]=\left[x^{2} + \frac{2}{3}\right]-1=k\left(let\right)$
Now, the function is $sin^{- 1}k-cos^{- 1}\left(k + 1\right)$
For the above function to be defined,
$-1\leq k\leq 1\&-1\leq \left(k + 1\right)\leq 1$
$\therefore k=\left\{- 1,0\right\}$
$\therefore $ Range $=\left\{s i n^{- 1} \left(- 1\right) - c o s^{- 1} \left(0\right) , s i n^{- 1} \left(0\right) - c o s^{- 1} \left(1\right)\right\}$ $=\left\{- \pi , 0\right\}$