Question

Find the domain of the following function:

$f\left(x\right)=\frac{1}{\left[x\right]}+{\log }_{1-x}(x^2-3x+10)+\frac{1}{2-|x|}+\frac{1}{\sqrt{\sec \left(\sin x\right)}}$

• A. $(-2,-1)\cup (-1,0)\cup (1,2)$
• B. $(-2,-1)\cup (-1,0)$
• C. $(-2,-1)\cup (1,2)$
• D. $(-1,0)\cup (1,2)$

Answer 1

Answer: (B)

$(-2,-1)\cup (-1,0)$

$\frac{1}{\left[x\right]}\Rightarrow x\notin [0,1)$

and ${\log }_{1-x}(x^2-3x+10)$

$x^2-3x+10>\Rightarrow x\in R$

$1-x>0\Rightarrow x\in R$

$2-x\ne 1\Rightarrow x\notin 1$

and $2-\mid x\mid =0\Rightarrow \mid x\mid -2=0$

$\Rightarrow x\ne -2,2$

and $\sec \left(\sin x\right)>0\Rightarrow -1\le \sin x\le 1$

$\Rightarrow x\in R$

Hence domain is $x\in (-2,-1)\cup (-1,0)$