Question

The range of the function $f\left(x\right)=\sqrt{x-1}+2\sqrt{3-x}$ is

• A. $[\sqrt{2},\sqrt{8}]$
• B. $[\sqrt{2},2]$
• C. $[2\sqrt{2},\sqrt{10}]$
• D. $[1,3]$

Answer 1

The correct option is A $[\sqrt{2},\sqrt{8}]$

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

$x-1\ge 0\Rightarrow x\ge 1$ ...(1)

and $3-x\ge 0\Rightarrow x\le 3$ ...(2)

From (1) and (2)

Domain of $f\left(x\right)$ is $[1,3]$

Then $f\left(1\right)=\sqrt{1-1}+2\sqrt{3-1}=0+2\sqrt{2}=2\sqrt{2}$

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

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