Let `f:[pi,3pi//2] to R ` be a function given by `f(x)=[sinx]+[1+sinx]+[2+ sinx]`
Then , the range of f(x) is

Options

A. `{0,3}`
B. `{1}`
C. `{0,2}`
D. `{3}`

Answer

To find the range of the function \( f(x) = [\sin x] + [1 + \sin x] + [2 + \sin x] \) where \( f: [\pi, \frac{3\pi}{2}] \to \mathbb{R} \), we will analyze the behavior of \( \sin x \) within the given interval.

### Step-by-Step Solution:

1. **Determine the range of \( \sin x \) on the interval \([\pi, \frac{3\pi}{2}]\)**:
- At \( x = \pi \), \( \sin(\pi) = 0 \).
- At \( x = \frac{3\pi}{2} \), \( \sin\left(\frac{3\pi}{2}\right) = -1 \).
- Therefore, as \( x \) varies from \( \pi \) to \( \frac{3\pi}{2} \), \( \sin x \) decreases from \( 0 \) to \( -1 \).

2. **Evaluate \( [\sin x] \)**:
- The greatest integer function \( [\sin x] \) will take values:
- \( [\sin x] = 0 \) when \( \sin x = 0 \) (at \( x = \pi \)).
- \( [\sin x] = -1 \) when \( -1 < \sin x < 0 \) (for \( x \) in \( (\pi, \frac{3\pi}{2}) \)).
- Thus, \( [\sin x] \) can be \( 0 \) or \( -1 \).

3. **Evaluate \( [1 + \sin x] \)**:
- Since \( \sin x \) ranges from \( 0 \) to \( -1 \), \( 1 + \sin x \) ranges from \( 1 \) to \( 0 \).
- Therefore:
- \( [1 + \sin x] = 1 \) when \( \sin x = 0 \) (at \( x = \pi \)).
- \( [1 + \sin x] = 0 \) when \( \sin x \) is in \( (-1, 0) \) (for \( x \) in \( (\pi, \frac{3\pi}{2}) \)).

4. **Evaluate \( [2 + \sin x] \)**:
- The term \( 2 + \sin x \) will range from \( 2 \) to \( 1 \).
- Therefore:
- \( [2 + \sin x] = 2 \) when \( \sin x = 0 \) (at \( x = \pi \)).
- \( [2 + \sin x] = 1 \) when \( \sin x \) is in \( (-1, 0) \) (for \( x \) in \( (\pi, \frac{3\pi}{2}) \)).

5. **Combine the values to find \( f(x) \)**:
- At \( x = \pi \):
\[ f(\pi) = [0] + [1] + [2] = 0 + 1 + 2 = 3 \]
- For \( x \) in \( (\pi, \frac{3\pi}{2}) \):
\[ f(x) = [-1] + [0] + [1] = -1 + 0 + 1 = 0 \]

6. **Conclusion**:
- The function \( f(x) \) takes the values \( 3 \) at \( x = \pi \) and \( 0 \) for \( x \) in \( (\pi, \frac{3\pi}{2}) \).
- Therefore, the range of \( f(x) \) is \( \{0, 3\} \).

### Final Answer:
The range of \( f(x) \) is \( \{0, 3\} \).