Question

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

• A. $\left[0,2\right]$
• B. $[0,2)$
• C. $[1,2)$
• D. $\left[1,2\right]$

Answer 1

The correct option is C

$[1,2)$

Explanation for correct answer Option(s)

Finding the domain of the function $f\left(x\right)$

The given function is defined as:

Case 1:

$$\begin{gathered} \Rightarrow 4-x^2\ge 0 \\ \Rightarrow x^2-4\le 0 \\ \Rightarrow \left(x-2\right)\left(x+2\right)\le 0 \\ \Rightarrow x\in \left[-2,2\right] \end{gathered}$$

Case 2:

${\sin }^{-1}\left(2-x\right)\ne 0$

$$\begin{gathered} \Rightarrow -1\le {\sin }^{-1}\left(2-x\right)\le 1 \\ \Rightarrow -1\le \left(2-x\right)\le 1 \\ \Rightarrow -1-2\le -x\le 1-2 \\ \Rightarrow -3\le -x\le -1 \\ \Rightarrow 3\ge x\ge 1 \\ \Rightarrow x\in \left[1,3\right]-\left\{2\right\}\text{or}[1,2)\cup (2,3] \end{gathered}$$

So, the domain of the given function is $[1,2)$

Therefore, Option C is correct.