Question

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

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

Answer 1

Answer: (B)

$(-\infty ,-1]$

$\log (1-x)$ is defined if $1-x>0\Rightarrow x<1\Rightarrow xϵ(-\infty ,1).....\left(1\right)$
$\sqrt{x^2-1}$ is defined if $x^2-1\ge 0\Rightarrow x^2\ge 1$
$\Rightarrow x\ge 1$ or $x\le -1.....\left(2\right)$
$\therefore$ Required domain is the instruction of $\left(1\right)$ and $\left(2\right)$
i.e. $(-\infty ,-1]$