Question

Find the domain of definition of the following function:
$y=\sqrt{1-\log (x-1)}+\sqrt{\frac{4-x}{x+2}}$

Answer 1

For $\sqrt{1-\log (x-1)}$

$1-\log (x-1)\ge 0$ and $x-1>0$

$1\ge \log (x-1)$ and $x>1$

${10}^1\ge x-111\ge xx\in (1,11]$

For $\sqrt{\frac{4-x}{x+2}}$

$\frac{4-x}{x+2}\ge 0(x+2)(4-x)\ge 0(x-4)(x+2)\le 0x\in (-2,4]x\ne -2$

Taking intersection we get

$x\in (1,4]$