Question

The domain of $f\left(x\right)={\log }_{[x+\frac{1}{2}]}|x^2-x-2|$ is ( where $[.]$ represents the greatest integer function)

• A. $(\frac{1}{2},\infty )$
• B. $(\frac{1}{2},\infty )-\left\{2\right\}$
• C. $(\frac{3}{2},\infty )$
• D. $[\frac{3}{2},2)\cup (2,\infty )$

Answer 1

The correct option is D $[\frac{3}{2},2)\cup (2,\infty )$

For $f\left(x\right)$ to be defined
$\left(i\right)[x+\frac{1}{2}]>0,[x+\frac{1}{2}]\ne 1$
$\Rightarrow [x+\frac{1}{2}]\ge 2$
$\Rightarrow x+\frac{1}{2}\ge 2,$
$\Rightarrow x\ge \frac{3}{2}$

$\left(ii\right)|x^2-x-2|>0$
$\Rightarrow |(x+1)(x-2)|>0$
$\Rightarrow x=R-\{-1,2\}$

$\therefore x\in [\frac{3}{2},2)\cup (2,\infty )$