Question

The domain of the function $f\left(x\right)=\frac{1}{lo{g}_{{}_{10}}\left(1-x\right)}+\sqrt{x+2}$ is

• A. $(-3,-2.5)\cap (2.5,-2)$
• B. $[-2,0)\cup (0,1)$
• C. $(0,1)$
• D. None of the above

Answer 1

Answer: (B)

$x+2\ge 0,$ ie., $x\ge -2$ or $-2\le x$
$\because {\log }_{10}(1-x)\ne 0$
$\Rightarrow 1-x\ne 1\Rightarrow x\ne 0$
Again, $1-x>0$
$\Rightarrow 1>x\Rightarrow x<1$
All these can be combined as
$-2\le x<0$ and $0<x<1$