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

JEE Main · 2026 · 06 Apr 2026 · Shift 1 · Sets Relations and Functions

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

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

[α, β)

, then

α^{2} + β^{2}

is equal to:

Answer: (B)

For

sin^{- 1} t

to be defined,

- 1 \leq t \leq 1

, so

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

. On the interval

[x] = n

with

x \in [n, n + 1)

x + [x] \in [2 n, 2 n + 1)

; this lies within

[- 3, 3]

only for

n = - 1, 0, 1

. These contribute

x \in [- 1, 0) \cup [0, 1) \cup [1, 2) = [- 1, 2)

. Hence

α = - 1

β = 2

and

α^{2} + β^{2} = 1 + 4 = 5

JEE Main · 2026 · 06 Apr 2026 · Shift 1 · Single correct option