Question

Find the domain of definition of the following function:
$y=\sqrt{{\log }_{x-2}(x^2-8x+15)}.$

Answer 1

${\log }_{x-2}(x^2-8x+15)\ge 0x\ne 2x^2-8x+15\ge 1x\in (-\infty ,-\sqrt{2}+4)\cup (4+\sqrt{2},\infty )$

and

$x-2>1x>3x\in (3,\infty )$

and

$0<x^2-8x+15\le 1$ and $0<x-2<1$

$(x-3)(x-5)>0$ and $x^2-8x+14\le 0$

$x\in (-\infty ,3)\cup (3,\infty )$ $x\in (4-\sqrt{2},4+\sqrt{2})$

$x>2$ and $x<3$

$x\in (2,3)$

Taking intersection, we get

$x\in (4-\sqrt{2},3)\cup (4+\sqrt{2},\infty )$