Question

Let $f : X \rightarrow Y$ 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 $f(3)=2$
$\therefore \quad$ Range $[\sqrt{2}, 2]$

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