If [x] denotes the integral part of x, then the domain of $f(x)=\cos^{-1}(x+[x])$ is

[0, 1)

For f(x) to be defined, $-1 ≤ x + |x| ≤ 1$

When $x∈I$, let x = k, then from (i), $-1 ≤ 2k ≤ 1$

$⇒ -\frac{1}{2}≤k≤\frac{1}{2}⇒ k = 0 ⇒x =0$   …(i)

When $x ∉ I$, let $x = k + α$, where k is the integral part of x and $0 < α < 1$

From (i), $−1≤ 2k + α ≤1 ⇒ \frac{-1-α}{2}≤k≤\frac{1-α}{2}⇒ k = 0 < 0 ⇒ x <1$   …(ii)

From (i) and (ii), all possible values of x are given by 0 ≤ x < 1

∴ Domain of f = [0, 1)