Question

If $[.]$ denotes the greatest integer function then the domain of the real-valued function ${\log }_{[x+1/2]}|x^2-x-2|$ is

• A. $x\in [\frac{3}{2},\infty )-\left\{2\right\}$
• B. $[\frac{5}{2},+\infty )$
• C. $(\frac{1}{2},2)\cup (2,+\infty )$
• D. none of these

Answer 1

The correct option is A $x\in [\frac{3}{2},\infty )-\left\{2\right\}$

Let $f\left(x\right)={\log }_{[x+1/2]}|x^2-x-2|$

For $f\left(x\right)$ to be defined $x^2-x-2\ne 0\Rightarrow (x-2)(x+1)\ne 0\Rightarrow x\ne -1,2$

And $[x+\frac{1}{2}]>0,\ne 1\Rightarrow x+\frac{1}{2}\ge 1,$ and $x+\frac{1}{2}\notin [1,2)$

$\Rightarrow x\ge \frac{1}{2}$ and $x\notin [\frac{1}{2},\frac{3}{2})$

Combining above we get required domain $x\in [\frac{3}{2},\infty )-\left\{2\right\}$