Question

The domain of the function $f\left(x\right)=\frac{\sqrt{{\log }_{0.4}\left(\frac{x-1}{x+5}\right)}}{x^2-36}$ is

• A. $(1,\infty )$
• B. $(6,\infty )$
• C. $(-5,\infty )-\left\{6\right\}$
• D. $(1,\infty )-\left\{6\right\}$

Answer 1

The correct option is D $(1,\infty )-\left\{6\right\}$

For domain of $f\left(x\right)$,
$\left(i\right){\log }_{0.4}\left(\frac{x-1}{x+5}\right)\ge 0\Rightarrow \frac{x-1}{x+5}\le 1$
$\Rightarrow \frac{-6}{x+5}\le 0\Rightarrow x>-5$

$\left(ii\right)\frac{x-1}{x+5}>0\Rightarrow x\in (-\infty ,-5)\cup (1,\infty )$

$\left(iii\right)x^2-36\ne 0\Rightarrow x\ne \pm 6$
$\therefore$ Domain $=(1,\infty )-\left\{6\right\}$