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  4. If f(x)= log e(6-|x2+x-6|), then domain of f(x) has how many integral values of x ?

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Q. 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 $x$ ?

ManipalManipal 2020

Solution:

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 $x^{2}+x-6 < 6$
$\Rightarrow x^{2}+x-12 < 0$
$\Rightarrow(x+4)(x-3) < 0$
image
$\therefore x \in(-4,3)$ ...(i)
Now, if $-6 \Rightarrow x^{2}+x > 0$
$\Rightarrow x(x+1) > 0$
image
$\therefore x \in(0, \infty)$ ...(ii)
From Eqs. (i) and (ii), we get
$x \in(0,3)$
$\because f(x)$ has only two integral values.
$\therefore x=1,2$