`f(x)=sin^(-1)((2-3[x])/4)`, which `[*]` denotes the greatest integer function.

Answer

To find the domain of the function \( f(x) = \sin^{-1}\left(\frac{2 - 3[x]}{4}\right) \), where \([x]\) denotes the greatest integer function, we need to ensure that the argument of the inverse sine function lies within the valid range of \([-1, 1]\).

### Step-by-Step Solution:

1. **Set up the inequality for the inverse sine function:**
\[ -1 \leq \frac{2 - 3[x]}{4} \leq 1 \]

2. **Multiply all parts of the inequality by 4:**
\[ -4 \leq 2 - 3[x] \leq 4 \]

3. **Rearranging the left side of the inequality:**
\[ -4 - 2 \leq -3[x] \implies -6 \leq -3[x] \]
Dividing by -3 (and reversing the inequality):
\[ 2 \geq [x] \]
This gives us:
\[ [x] \leq 2 \]

4. **Rearranging the right side of the inequality:**
\[ 2 - 3[x] \leq 4 \implies -3[x] \leq 2 \implies [x] \geq -\frac{2}{3} \]
Since \([x]\) is an integer, this means:
\[ [x] \geq 0 \]

5. **Combining the results from both inequalities:**
We have:
\[ 0 \leq [x] \leq 2 \]
This means \([x]\) can take values \(0, 1, 2\).

6. **Finding the corresponding values of \(x\):**
- If \([x] = 0\), then \(0 \leq x < 1\).
- If \([x] = 1\), then \(1 \leq x < 2\).
- If \([x] = 2\), then \(2 \leq x < 3\).

7. **Combining these intervals:**
The combined interval for \(x\) is:
\[ 0 \leq x < 3 \]

### Final Domain:
Thus, the domain of the function \( f(x) \) is:
\[ \text{Domain of } f(x) = [0, 3) \]