If the domain of the function $f(x)=\cos ^{-1}\left(\frac{2-|x|}{4}\right)+\left(\log _{e}(3-x)\right)^{-1}$ is $[-\alpha, \beta)-\{\gamma\}$, then $\alpha+\beta+\gamma$ is equal to :

JEE Main · 2024 · 30 Jan 2024 · Shift 1 · Inverse Trigonometric Functions

If the domain of the function

f (x) = cos^{- 1} (\frac{2 - ∣ x ∣}{4}) + (lo g_{e} (3 - x))^{- 1}

[- α, β) - {γ}

, then

α + β + γ

is equal to :

Answer: (C)

cos^{- 1}

needs

- 1 \leq \frac{2 - ∣ x ∣}{4} \leq 1 \Rightarrow - 4 \leq 2 - ∣ x ∣ \leq 4 \Rightarrow - 2 \leq ∣ x ∣ \leq 6 \Rightarrow x \in [- 6, 6]

.

The log term needs

3 - x > 0

and

3 - x \neq = 1

, i.e.

x < 3, x \neq = 2

.

Combined domain:

[- 6, 3) - {2}

, so

α = 6, β = 3, γ = 2

, and

α + β + γ = 11

JEE Main · 2024 · 30 Jan 2024 · Shift 1 · Single correct option

If the domain of the function $f (x) = cos^{- 1} (\frac{2 x - 5}{11 - 3 x}) + sin^{- 1} (2 x^{2} - 3 x + 1)$ is the interval $[α, β]$ , then $α + 2 β$ is equal to :
JEE Main 2026View solution →
The number of solutions of $tan^{- 1} 4 x + tan^{- 1} 6 x = \frac{π}{6}$ , where $- \frac{1}{2 6} < x < \frac{1}{2 6}$ is equal to
JEE Main 2026View solution →
If the domain of the function $f (x) = sin^{- 1} (\frac{1}{x ^{2} - 2 x - 2})$ , is $(- \infty, α] \cup [β, γ] \cup [δ, \infty)$ , then $α + β + γ + δ$ is equal to
JEE Main 2026View solution →
If $k = tan (\frac{π}{4} + \frac{1}{2} cos^{- 1} (\frac{2}{3})) + tan (\frac{1}{2} sin^{- 1} (\frac{2}{3}))$ then the number of solutions of the equation …
JEE Main 2026View solution →
Considering the principal values of inverse trigonometric functions, the value of the expression $tan (2 sin^{- 1} (\frac{2}{13}) - 2 cos^{- 1} (\frac{3}{10}))$ is equal to :
JEE Main 2026View solution →
Let [ • ] denote the greatest integer function. If the domain of the function $f (x) = cos^{- 1} (\frac{4 x + 2 [ x ]}{3})$ is $[α, β]$ , then $12 (α + β)$ is equal to :
JEE Main 2026View solution →