Functions Question 8
Question 8 - 2024 (30 Jan Shift 1)
If the domain of the function $f(x)=\cos ^{-1}\left(\frac{2-|x|}{4}\right)+\left(\log _{e}(3-x)\right)^{-1}$ is $[-\alpha, \beta)-{y}$, then $\alpha+\beta+\gamma$ is equal to :
(1) 12
(2) 9
(3) 11
(4) 8
Show Answer
Answer (3)
Solution
$-1 \leq\left|\frac{2-|x|}{4}\right| \leq 1$
$\Rightarrow\left|\frac{2-|x|}{4}\right| \leq 1$
$-4 \leq 2-|x| \leq 4$
$-6 \leq-|x| \leq 2$
$-2 \leq|x| \leq 6$
$|x| \leq 6$
$\Rightarrow x \in[-6,6]$
Now, $3-x \neq 1$
And $x \neq 2$…(2)
and $3-x>0$
$x<3$
From (1), (2) and (3)
$\Rightarrow \quad x \in[-6,3)-{2}$
$\alpha=6$
$\beta=3$
$\gamma=2$
$\alpha+\beta+\gamma=11$
