Question

The domain of the function $f \left(x\right)=\frac{1}{log_{_{10}}\left(1-x\right)}+\sqrt{x+2}$ 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 \geq 0, $ ie., $ x \geq-2 $ or $-2 \leq x $
$ \because \log _{10}(1-x) \neq 0 $
$ \Rightarrow 1-x \neq 1 \Rightarrow x \neq 0 $
Again, $1-x > 0 $
$ \Rightarrow 1>x \Rightarrow x < 1$
All these can be combined as
$-2 \leq x < 0$ and $ 0 < x < 1$