Question

The domain of the function $f\left(x\right)=\log (1-x)+\sqrt{x^2-1}$ is

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

Answer 1

The correct option is B $(-\infty ,-1]$

$\log (1-x)$ is defined if $1-x>0\Rightarrow x<1$
$\Rightarrow x\in (-\infty ,1)\cdots \left(1\right)$
$\sqrt{x^2-1}$ is defined if $x^2-1\ge 0\Rightarrow x^2\ge 1$
$\Rightarrow x\ge 1$ or $x\le -1\cdots \left(2\right)$
From $\left(1\right)$ and $\left(2\right),$ we get
Domain as $(-\infty ,-1]$