Question

Find the domain of definition and the range of the following function:
$f\left(x\right)=\sqrt{x-1}+2\sqrt{3-x}$

Answer 1

$f\left(x\right)=\sqrt{x-1}+2\sqrt{3-x}$

$x-1\ge 0$

$x\ge 1$

$3-x\ge 0$

$x\le 3$

So the Domain of $f\left(x\right)=[1,3]$

When $x=1$

$f\left(x\right)=\sqrt{x-1}+2\sqrt{3-x}$

$=\sqrt{1-1}+2\sqrt{3-1}$

$=0+2\sqrt{2}=2\sqrt{2}$

When $x=3$

$f\left(x\right)=\sqrt{3-1}+2\sqrt{3-3}$

$=\sqrt{2}$

The range of $f\left(x\right)=[\sqrt{2},2\sqrt{2}]$