Find the domain and range of `f(x)=log[ cos|x|+1/2]`,where [.] denotes the greatest integer function.

Answer

To find the domain and range of the function \( f(x) = \log\left[\cos|x| + \frac{1}{2}\right] \), where \([.]\) denotes the greatest integer function, we will proceed step by step.

### Step 1: Determine the condition for the logarithm to be defined
The logarithm function is defined only for positive values. Therefore, we need:
\[ \cos|x| + \frac{1}{2} > 0 \]
This implies:
\[ \cos|x| > -\frac{1}{2} \]

### Step 2: Analyze the cosine function
The cosine function, \(\cos|x|\), oscillates between -1 and 1. We need to find the values of \(x\) for which \(\cos|x| > -\frac{1}{2}\).

### Step 3: Find the intervals for \(\cos|x| > -\frac{1}{2}\)
The cosine function equals \(-\frac{1}{2}\) at specific angles:
\[ |x| = \frac{2\pi}{3} + 2k\pi \quad \text{and} \quad |x| = \frac{4\pi}{3} + 2k\pi \quad \text{for } k \in \mathbb{Z} \]
This means:
- \(\cos|x| = -\frac{1}{2}\) at \(x = \pm \frac{2\pi}{3} + 2k\pi\) and \(x = \pm \frac{4\pi}{3} + 2k\pi\).

### Step 4: Determine the intervals
The intervals where \(\cos|x| > -\frac{1}{2}\) can be determined from the unit circle:
- Between \(-\frac{2\pi}{3}\) and \(\frac{2\pi}{3}\), and between \(-\frac{4\pi}{3}\) and \(\frac{4\pi}{3}\), and so on.

Thus, the general solution for the domain can be expressed as:
\[ x \in \left(2n\pi - \frac{2\pi}{3}, 2n\pi + \frac{2\pi}{3}\right) \quad \text{for } n \in \mathbb{Z} \]

### Step 5: Determine the range of the function
Next, we need to analyze the range of \(f(x)\):
\[ f(x) = \log\left[\cos|x| + \frac{1}{2}\right] \]
The maximum value of \(\cos|x|\) is 1, so:
\[ \cos|x| + \frac{1}{2} \leq 1 + \frac{1}{2} = \frac{3}{2} \]
Thus:
\[ \cos|x| + \frac{1}{2} \geq 0 \quad \text{(since we established the domain)} \]
The minimum value occurs when \(\cos|x| = -\frac{1}{2}\):
\[ \cos|x| + \frac{1}{2} \geq 0 \implies \text{minimum value is } 0 \]
The greatest integer function \([ \cdot ]\) will yield:
\[ 1 \leq \cos|x| + \frac{1}{2} < \frac{3}{2} \implies [\cos|x| + \frac{1}{2}] = 1 \]
Thus:
\[ f(x) = \log(1) = 0 \]

### Conclusion
The domain of \(f(x)\) is:
\[ x \in \left(2n\pi - \frac{2\pi}{3}, 2n\pi + \frac{2\pi}{3}\right) \quad \text{for } n \in \mathbb{Z} \]
The range of \(f(x)\) is:
\[ \{0\} \]