Question

Let $f:X\to Y$ be a function such that $f(x)=\sqrt{x-2}+\sqrt{4-x}$, then the set of $X$ and $Y$ for which $f(x)$ is both injective as well as surjective, is

• A. $[2,4]$ and $[\sqrt{2},2]$
• B. $[3,4]$ and $[\sqrt{2},2]$
• C. $[2,4]$ and $[1,2]$
• D. $[2,3]$ and $[1,2]$

Answer 1

Answer: (B)

Clearly domain is $x\in [2,4]$
Now, $f^{\prime }(x)=0\Rightarrow x=3$ and $f(3)=2$
$\therefore$ Range $[\sqrt{2},2]$

But, domain has to be restricted in either $[2,3]$ or $[3,4]$ for function to be invertible.