Domain of $\cos^{-1}[x]$ is, where $[.]$ denotes greatest integer function:
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The greatest integer function $[x] = n$ implies $n \le x <n+1$. Always remember that the upper bound in the domain of $[x]$ results in an open bracket ($2$ is not included because $[2]=2$, which is outside the range $[-1, 1]$).