Question

Determine domain of the function $f\left(x\right)=\sqrt{{\log }_{a4}(x-x^2)}$, the domain is (0,k), k is?

Answer 1

${\log }_{a4}(x-x^2)\ge 0={\log }_{a4}1$
Since base is less than 1 above relation implies that $x-x^2<1$or $0<x^2-x+1$ or $x^2-x+1>0$or ${(x-\frac{1}{2})}^2+1-\frac{1}{4}>0$ or ${(x-\frac{1}{2})}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2>0$
Above is true for all $xϵR$Also $x-x^2\ge 0$ by definition of ${\log }_{a}z$ or $x(1-x)\ge 0$ or $x(x-1)\le 0$
$\therefore xϵ(0,1)\therefore xϵ(0,1)\cap R=(0,1)$