If the domain of the function is then is equal to

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Solution

Q.
If the domain of the function $f (x) = \cos^{- 1} (\frac{2 - | x |}{4}) + {\log_{e} (3 - x)}^{- 1}$ is $[- α, β) - {γ},$ then $α + β + γ$ is equal to [2024]

1 9

2 8

3 11

4 12

Ans.
(3)

We have, $f (x) = \cos^{- 1} (\frac{2 - | x |}{4}) + {\log_{e} (3 - x)}^{- 1}$

For $f (x)$ be defined $- 1 \leq \frac{2 - | x |}{4} \leq 1$

$\Rightarrow - 4 \leq 2 - | x | \leq 4 \Rightarrow - 6 \leq - | x | \leq 2$

$\Rightarrow 6 \geq | x | \geq - 2$

Since, $| x | \geq - 2$ so $- 6 \leq x \leq 6$ ...(i)

Also, $3 - x > 0$ and $3 - x \neq 1$

$\Rightarrow x < 3 and x \neq 2$ ...(ii)

Taking intersection of (i) and (ii), we get $x \in [- 6, 3) - {2}$

$\Rightarrow α = 6, β = 3, and γ = 2$

$\Rightarrow α + β + γ = 6 + 3 + 2 = 11$

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