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
• Correct answer
  D

  None of these

Solution

Given, $f(x)=\log _e\left(6-\left|x^2+x-6\right|\right)$ The function $f(x)$ is defined, if $\begin{array}{lr} & \left(6-\left|x^2+x-6\right|\right)\gt0 \\ \Rightarrow & \left|x^2+x-6\right| \lt 6 \\ \Rightarrow & -6 \lt x^2+x-6 \lt 6 \\ \text { If } & x^2+x-6 \lt 6 \\ \Rightarrow & x^2+x-12 \lt 0 \\ \Rightarrow & (x+4)(x-3) \lt 0\end{array}$
$\therefore \quad x \in(-4,3)$ ....(i) Now, if $-6 \lt x^2+x-6$ $\Rightarrow \quad x^2+x\gt0 \Rightarrow x(x+1)\gt0$
From Eqs. (i) and (ii), we get $x \in(0,3)$ $\because f(x)$ has only two integral values. $\therefore \quad x=1,2$

Hence, the correct answer is option: 'D'.