Example 10. Find the domain of the function

$$
y=\arcsin \frac{2 x-1}{3}
$$
Solution. The function $u=\arcsin t$ is defined for $-1 \leq t \leq 1$. Therefore, the given function is defined only for those values of $x$ for which

$$
-1 \leq \frac{2 x-1}{3} \leq 1 \text {, from which }-3 \leq 2 x-1 \leq 3
$$

Solving these inequalities, we get $-1 \leq x \leq 2$. Thus, the function is defined on the interval $[-1,2]$.