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