Let [ • ] denote the greatest integer function. If the domain of the function $f(x)=\cos ^{-1}\left(\frac{4 x+2[x]}{3}\right)$ is $[\alpha, \beta]$, then $12(\alpha+\beta)$ is equal to :

JEE Main · 2026 · 04 Apr 2026 · Shift 1 · Inverse Trigonometric Functions

Let [ • ] denote the greatest integer function. If the domain of the function

f (x) = cos^{- 1} (\frac{4 x + 2 [ x ]}{3})

[α, β]

, then

12 (α + β)

is equal to :

Answer: (A)

The function needs

- 1 \leq \frac{4 x + 2 [ x ]}{3} \leq 1

, i.e.

- 3 \leq 4 x + 2 [x] \leq 3

. On

0 \leq x < 1

[x] = 0

gives

- 3 \leq 4 x \leq 3

, so

0 \leq x \leq \frac{3}{4}

. On

- 1 \leq x < 0

[x] = - 1

gives

- 3 \leq 4 x - 2 \leq 3

, so

- \frac{1}{4} \leq x < 0

. Every other integer interval gives an empty intersection. The domain is

[- \frac{1}{4}, \frac{3}{4}]

, so

α = - \frac{1}{4}

β = \frac{3}{4}

and

12 (α + β) = 12 \cdot \frac{1}{2} = 6

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