Question

Find the domain of definition of the following function:
$y=\sqrt{{\log }_{0.5}(x^2-9)+4}$

Answer 1

$y=\sqrt{{\log }_{0.5}(x^2-9)+4}$

$⟹x^2-9>0⟹x\in (-\infty ,-3)\cup (3,\infty )$

And ${\log }_{0.5}(x^2-9)+4\ge 0⟹x^2-9\le (0.5{)}^{-4}⟹x\in [-\sqrt{3.0625},\sqrt{3.0625}]$

So $x\in [-\sqrt{3.0625},3)\cup (3,\sqrt{3.0625})$