The range of the function defined as `f(x) = log_(3)((1)/(sqrt([cosx] - [sin x])))`, where [ ] represents the greatest integer function, is

Options

A. {0}
B. `[-(pi)/(2), 0]`
C. [0, 1]
D. {1}

Answer

To find the range of the function \( f(x) = \log_3\left(\frac{1}{\sqrt{[\cos x] - [\sin x]}}\right) \), where \([\cdot]\) denotes the greatest integer function, we will proceed step by step.

### Step 1: Understand the greatest integer function
The greatest integer function \([y]\) gives the largest integer less than or equal to \(y\). Therefore, we need to evaluate \([\cos x]\) and \([\sin x]\).

### Step 2: Determine the ranges of \(\cos x\) and \(\sin x\)
The values of \(\cos x\) and \(\sin x\) oscillate between -1 and 1. Hence, we can have the following cases for \([\cos x]\) and \([\sin x]\):
- \([\cos x] = 0\) when \(0 \leq \cos x < 1\)
- \([\cos x] = -1\) when \(-1 \leq \cos x < 0\)
- \([\sin x] = 0\) when \(0 \leq \sin x < 1\)
- \([\sin x] = -1\) when \(-1 \leq \sin x < 0\)

### Step 3: Evaluate \([\cos x] - [\sin x]\)
Now we will analyze the possible values of \([\cos x] - [\sin x]\):
1. If \([\cos x] = 0\) and \([\sin x] = 0\), then \([\cos x] - [\sin x] = 0\).
2. If \([\cos x] = 0\) and \([\sin x] = -1\), then \([\cos x] - [\sin x] = 1\).
3. If \([\cos x] = -1\) and \([\sin x] = 0\), then \([\cos x] - [\sin x] = -1\).
4. If \([\cos x] = -1\) and \([\sin x] = -1\), then \([\cos x] - [\sin x] = 0\).

### Step 4: Identify valid cases
For the logarithm to be defined, we need \([\cos x] - [\sin x] > 0\):
- The only valid case is when \([\cos x] = 0\) and \([\sin x] = -1\), which gives \([\cos x] - [\sin x] = 1\).

### Step 5: Substitute into the function
Now substituting this into the function:
\[ f(x) = \log_3\left(\frac{1}{\sqrt{1}}\right) = \log_3(1) = 0 \]

### Step 6: Conclusion
Thus, the range of the function \( f(x) \) is:
\[ \text{Range} = \{0\} \]