Question

The domain of the function $f\left(x\right)={\log }_3{\log }_{1/3}(x^2+10x+25)+\frac{1}{\left[x\right]+5}$ where [.] denotes the greatest integer function) is

• A. $(-4,-3)$
• B. $(-6,-5)$
• C. $(-6,4)$
• D. None of these

Answer 1

The correct option is B $(-6,-5)$

For given function to be defined,
(i) $x^2+10x+25>0\Rightarrow (x+5{)}^2>0\Rightarrow x\ne -5=D_1$
(ii) ${\log }_{1/3}(x^2+10x+25)>0\Rightarrow x^2+10x+25<1$
$\Rightarrow x^2+10x+24<0\Rightarrow (x+4)(x+6)<0\Rightarrow x\in (-6,-4)=D_2$
and
(iii) $\left[x\right]+5\ne 0\Rightarrow x\notin [-5,4)=D_3$
Hence domain of f is $D_{!}\cap D_2\cap D_3=(-6,-5)$