Question

Find the domain of definition of the following function:
$y=\sqrt{{\log }_{\frac{1}{2}}\frac{x-1}{3x+5}}.$

Answer 1

$y=\sqrt{{log}_{\frac{1}{2}}\frac{x-1}{(3x+5)}}$

$\frac{x-1}{3x+5}>0$ for $y$ to be real and defined.

Therefore,$x-1>0$ and $3x+5>0$ or $x-1<0$ and $3x+5<0$

$x\in (-\infty ,\frac{-5}{3})\cup (1,\infty ){log}_{\frac{1}{2}}\frac{(x-1)}{(3x+5)}\ge 0\Rightarrow \frac{x-1}{3x+5}\le 1\Rightarrow \frac{-2x-6}{3x+5}\le 0\Rightarrow \frac{x+3}{3x+5}\ge 0$

Therefore, $x\in (-\infty ,3]\cup (\frac{5}{3},\infty )$

Therefore Domain of $y$ is $x\in (-\infty ,3]\cup (1,\infty ).$