Question

Let $f\left(x\right)=\sqrt{4-\sqrt{2-x}}$ and $g\left(x\right)=(x-a)(x-a+3).$ If $g\left(f\right(x\left)\right)<0\forall x\in D_f,$ then the complete set of values of $a$ is
$[D_f$ denotes the domain of the function $f]$

• A. $(2,5)$
• B. $(2,3)$
• C. $(0,3)$
• D. $(0,5)$

Answer 1

The correct option is B $(2,3)$

$f\left(x\right)=\sqrt{4-\sqrt{2-x}}$
$4-\sqrt{2-x}\ge 0$
$\Rightarrow 16\ge 2-x$
$\Rightarrow x\ge -14$
and $2-x\ge 0\Rightarrow x\le 2$
$\therefore D_f=[-14,2]$
Range of $f$ is $[0,2]$

$g\left(x\right)=(x-a)(x-a+3)$
$\Rightarrow g\left(f\right(x\left)\right)<0\forall f\left(x\right)\in [0,2]$
$\Rightarrow g\left(0\right)<0$ and $g\left(2\right)<0$
$\Rightarrow -a(-a+3)<0$ and $(2-a)(5-a)<0$
$\Rightarrow a\in (0,3)\cdots \left(1\right)$ and $a\in (2,5)\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right),$ we get
$a\in (2,3)$