Question

Find the domain of definition of the following function:
$y=\sqrt{\frac{-{\log }_{0.3}(x-1)}{\sqrt{-x^2+2x+8}}}$

Answer 1

$-{\log }_{0.3}(x-1)\ge 0{\log }_{0.3}(x-1)\le 0$

As the base is less than $1$

$x-1>1x>2$

Also, $-x^2+2x+8>0x^2-2x-8<0(x-4)(x+2)<0x\in (-2,4)$

Taking intersection , we get

$x\in (2,4)$