Question

If $f(x)={\log }_e\left(6-\left|x^2+x-6\right|\right)$, then domain of $f(x)$ has how many integral values of $x$ ?

• A. 5
• B. 4
• C. Infinite
• D. None of these

Answer 1

Answer: (D)

Given, $f(x)={\log }_e\left(6-\left|x^2+x-6\right|\right)$
The function $f(x)$ is defined, if
$\left(6-\left|x^2+x-6\right|\right)>0$
$\Rightarrow \left|x^2+x-6\right|<6$
$\Rightarrow -6<x^2+x-6<6$
if $x^2+x-6<6$
$\Rightarrow x^2+x-12<0$
$\Rightarrow (x+4)(x-3)<0$

$\therefore x\in (-4,3)$ ...(i)
Now, if $-6\Rightarrow x^2+x>0$
$\Rightarrow x(x+1)>0$

$\therefore x\in (0,\infty )$ ...(ii)
From Eqs. (i) and (ii), we get
$x\in (0,3)$
$\because f(x)$ has only two integral values.
$\therefore x=1,2$