Question

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

• A. $(-3,3)$
• B. $[-3,3]$
• C. $(-\infty ,-3)\cup (3,\infty )$
• D. $(-\infty ,-3]\cup [3,\infty )$

Answer 1

The correct option is B

$[-3,3]$

Explanation for the correct option:

We know that the domain of ${\cos }^{-1}x$ is $[-1,1]$

Therefore, we have the limits on the function within $\text{cosine}$ as:

$$\begin{gathered} \Rightarrow -1\le \frac{1-\left|x\right|}{2}\le 1 \\ \Rightarrow -2\le 1-\left|x\right|\le 2 \\ \Rightarrow -3\le -\left|x\right|\le 1...\left(1\right) \\ \Rightarrow 3\le \left|x\right|\le -1 \\ \Rightarrow -1\le \left|x\right|\le 3...\left(2\right) \end{gathered}$$

Now we know that,

for $\left|x\right|\ge 0$, we have :$\left|x\right|\le 3$, from $\left(2\right)$

$-3\le x\le 3$

Thus, we can conclude that the domain of the function is $[-3,3]$.

Hence, the correct answer is option (B).