53. $y=\arcsin \frac{x-2}{3}$.
Solution. The function is defined if

$$
\left\{\begin{array}{l}
\frac{x-2}{3} \geqslant-1 \\
\frac{x-2}{3} \leqslant 1
\end{array}\right.
$$

The solution to the system of inequalities are values $x \geqslant-1$ and $x \leqslant 5$. Therefore, the domain of the function is the interval $[-1,5]$.