Question

$\left\{ x \in \mathbb{R} \,\middle|\, \frac{\sqrt{|x^2 - 2|x| - 8}}{\log(2 - x - x^2)} \text{ is a real number} \right\} =$

• A. $(-\infty, -4] \cup [4, \infty)$
• B. $\varnothing$
• C. $(-1, 2)$
• D. $(-\infty, -4] \cup (-1, 2) \cup [4, \infty)$

Hint:

Always verify the domain restrictions imposed by both square roots and logarithmic expressions.

Answer 1

The Correct Option is B

We are given the expression: $\frac{\sqrt{|x^2 - 2|x| - 8}}{\log(2 - x - x^2)}$
To be real and defined: % Option (1) The quantity under the square root must be non-negative:
$\Rightarrow |x^2 - 2|x|| \geq 8$
% Option (2) The log function must be defined and positive:
$\Rightarrow 2 - x - x^2>0 \Rightarrow x^2 + x - 2<0 \Rightarrow x \in (-2, 1)$
We must also ensure the log argument $\neq 1$ to avoid division by zero.
On checking for common $x$ values that satisfy both conditions, we find no such values exist.
Hence, the domain is empty.