Discussion
The domain of \(f(x) = \sqrt{|x| - 1} + \sqrt{4 - |x|}\) is
Answer: E
The domain is \([-4, -1] \cup [1, 4]\).
Both square roots need non-negative arguments:
\[ \begin{aligned} |x| - 1 \ge 0 &\Rightarrow |x| \ge 1 \\ 4 - |x| \ge 0 &\Rightarrow |x| \le 4 \end{aligned} \]
So \(1 \le |x| \le 4\), which on the number line is \(-4 \le x \le -1\) or \(1 \le x \le 4\). All four endpoints are included, because a square root of zero is defined.
The options with round brackets wrongly exclude an endpoint such as \(x = 1\) (where \(f(1) = 0 + \sqrt{3}\)) or \(x = 4\). \([-4, -1]\) alone forgets the positive half.
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