Question

Find the domain of definition of the following function:
$\sqrt{{\log }_{\frac{1}{2}}\frac{x}{x^2-1}}$

Answer 1

first of all $x\ne \pm 1$

$lo{g}_{1/2}\frac{x}{x^2-1}\ge 0$

$⟹\frac{x}{x^2-1}>1$

$x>x^2-1$

$x^2-x-1<0$

$x\in (\frac{1-\sqrt{5}}{2},\frac{1+\sqrt{5}}{2})$

$⟹x\in (\frac{1-\sqrt{5}}{2},\frac{1+\sqrt{5}}{2})-\pm 1$