Question

Find the domain of $x$ if $f\left(x\right)=\sqrt{x^2-|x|-2}$

• A. $x\in R-(-2,2)$
• B. $x\in R$
• C. $x\in R-(0,2)$
• D. None of these

Answer 1

The correct option is A $x\in R-(-2,2)$

Case I $x<0$
Hence $f\left(x\right)=\sqrt{x^2-(-x)-2}$
$=\sqrt{x^2+x-2}$
$=\sqrt{(x+2)(x-1)}$
Therefore the domain will be $(-\infty ,-2]$....$\left(D_1\right)$
Case II $x>0$
Hence $f\left(x\right)=\sqrt{x^2-\left(x\right)-2}$
$=\sqrt{x^2-x-2}$
$=\sqrt{(x-2)(x+1)}$
Therefore the domain will be $[2,\infty )$....$\left(D_2\right)$
Hence the domain of the function $f\left(x\right)$ from $D_1$ and $D_2$ will be
$=(-\infty ,-2]\cup [2,\infty )$