Question

The domain of the function $f\left(x\right)=\sqrt{(x^2-5x+6)}+\sqrt{(8-x^2+2x)}$ is

• A. $[-2,4]$
• B. $[-2,2]\cup [3,4]$
• C. $[2,4]$
• D. $(-\infty ,2)$

Answer 1

The correct option is B $[-2,2]\cup [3,4]$

For function $f\left(x\right)$ to be defined
$x^2-5x+6\ge 0$
$\Rightarrow (x-2)(x-3)\ge 0$
$\Rightarrow x\in (-\infty ,2]\cup [3,\infty )\cdots \left(1\right)$
and
$-x^2+2x+8\ge 0$
$\Rightarrow x^2-2x-8\le 0$
$\Rightarrow (x-4)(x+2)\le 0$
$\Rightarrow x\in [-2,4]\cdots \left(2\right)$
From $\left(1\right)$ and $\left(2\right)$
$\therefore x\in [-2,2]\cup [3,4]$

Hence domain of the function is $[-2,2]\cup [3,4]$