Question

Find the domain of definition of the following function:
$y=\sqrt{-x^2+2x+3}+{\log }_3(x-1)$

Answer 1

$y=\sqrt{-x^2+2x+3}+{log}_{3}x-1$

For $4y$ to be real and defined,

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

$-x^2+2x+3\ge 0\Rightarrow x^2-2x-3\le 0\Rightarrow (x-3)(x+1)\le 0\Rightarrow x\in [-1,3]$

Therefore, domain of $y$ is $x\in (2,3].$