Question

The domain of the function $f(x) = \frac{\sqrt{ 4 - x^2}}{\sin^{-1} (2 - x)}$ is

• A. [0, 2]
• B. [0, 2)
• C. [1, 2)
• D. [1, 2]

Answer 1

Answer: (C)

$f\left(x\right) = \frac{\sqrt{4-x^{2}}}{\sin^{-1}\left(2-x\right)}$
$ \sqrt{4-x^{2}} $ is defined for $4 -x^{2} \ge0 $
$\Rightarrow x^{2} \le4$
$ \Rightarrow -2 \le x \le2$
and $ \sin^{-1} \left(2-x\right) $ is defined for $-1 \le2 - x \le1$
$\Rightarrow -3 \le - x \le-1$
$ \Rightarrow 1 \le x \le3$
Also, $ \sin^{-1} \left(2-x \right) = 0 $ for $x=2 $
$\therefore $ Domain of $f\left(x\right) = \left[-2 ,2\right]\cap\left[1,3\right] - \left\{2\right\}$
$ = [1, 2)$