Question

The domain of the function
$f\left(x\right)+{\sin }^{-1}[2x^2-3]+{\log }_2\left({\log }_{\frac{1}{2}}(x^2-5x+5)\right)$,
where $\left[t\right]$ is the greatest integer function, is

Answer 1

$–1\le 2x^2–3<2$ or $2\le 2x^2<5)$

$1\le x^2<\frac{5}{2}$

$x\in (-\sqrt{\frac{5}{2}},-1]$$\cup [1,\sqrt{\frac{5}{2}})$ ... eq.($1$)

$\Rightarrow {\log }_{\frac{1}{2}}(x^2-5x+5)>0$

$0<x^2-5x+5<1$

$x^2-5x+5>0$

$x\in (-\infty ,\frac{5-\sqrt{5}}{2})\cup (\frac{5+\sqrt{5}}{2},\infty )$... eq.$\left(2\right)$

$\Rightarrow x^2-5x+4<0$
$x\in (-\infty ,1)\cup (4,\infty )$. ...eq.$\left(3\right)$

Taking intersection from eq.(1),(2) and (3), we get
$x\in (1,\frac{5-\sqrt{5}}{2})$