Domain of $\operatorname{arccos}(x-1)$ All we know that domain of $\operatorname{arccos}(a)$ is $[-1; 1]$. So, if $a = x - 1$, then $-1 \leq x - 1 \leq 1$. I substracted one from all parts and got $-2\leq x\leq0$. But the answer is $[0;2]$. Where is mistake?
You have two inequalities:
$$
-1\le x-1 \quad \land \quad x-1\le1
$$
solve separately and you find:
$$
-1+1 \le x \quad \land \quad x\le 1+1
$$
The range of $arccos(x-1)$ is the same as the range of $arccosx$ which is $[0,\pi]$ It seems like you are confused with the domain of $arccos(x-1)$ which can be found by solving the inequality stated by others...