If the domain of `y=f(x)i s[-3,2],` then find the domain of `g(x)=f(|[x]|),w h e r[]` denotes the greatest integer function.

Explanation

To solve the problem, we need to find the domain of the function \( g(x) = f(|[x]|) \), given that the domain of \( f(x) \) is \([-3, 2]\).

### Step-by-Step Solution:

1. **Understand the Given Domain**: The domain of \( f(x) \) is given as \([-3, 2]\). This means that \( f(x) \) is defined for all \( x \) values between -3 and 2, inclusive.

2. **Analyze the Function \( g(x) \)**: The function \( g(x) = f(|[x]|) \) involves two operations:
- The greatest integer function \([x]\), which gives the largest integer less than or equal to \( x \).
- The absolute value function \(|[x]|\), which converts the output of the greatest integer function to a non-negative value.

3. **Determine the Range of \(|[x]|\)**:
- The greatest integer function \([x]\) can take any integer value. Therefore, \([x]\) can be any integer \( n \) such that \( n \leq x < n + 1 \).
- The absolute value \(|[x]|\) will yield non-negative integers, i.e., \( |[x]| \geq 0 \).

4. **Identify Values of \(|[x]|\) that Fit in the Domain of \( f(x) \)**:
- Since \( f(x) \) is defined for \( x \) in the interval \([-3, 2]\), we need to find the integer values of \(|[x]|\) that fall within this range.
- The possible values of \(|[x]|\) that are non-negative and within the domain of \( f(x) \) are \( 0, 1, 2 \).

5. **Find Corresponding Values of \( x \)**:
- For \(|[x]| = 0\): This occurs when \( [x] = 0 \), which means \( 0 \leq x < 1 \).
- For \(|[x]| = 1\): This occurs when \( [x] = 1 \), which means \( 1 \leq x < 2 \).
- For \(|[x]| = 2\): This occurs when \( [x] = 2 \), which means \( 2 \leq x < 3 \).

6. **Combine the Intervals**:
- From the above analysis, we have:
- From \(|[x]| = 0\): \( x \in [0, 1) \)
- From \(|[x]| = 1\): \( x \in [1, 2) \)
- From \(|[x]| = 2\): \( x \in [2, 3) \)
- Combining these intervals, we find that the domain of \( g(x) \) is \( [-2, 3) \).

### Final Result:
The domain of \( g(x) = f(|[x]|) \) is \( [-2, 3) \).