`f(x)=cos^-1sqrt(log_([x]) ((|x|)/x))` where [.] denotes the greatest integer function

Answer

To find the range of the function \( f(x) = \cos^{-1}\left(\sqrt{\log_{\lfloor x \rfloor}\left(\frac{|x|}{x}\right)}\right) \), we will first determine the domain of the function and then evaluate the range based on the domain.

### Step 1: Determine the Domain

1. **Understanding the logarithm**: The logarithm \( \log_a(b) \) is defined when:
- \( a > 0 \) and \( a \neq 1 \)
- \( b > 0 \)

In our case, \( a = \lfloor x \rfloor \) and \( b = \frac{|x|}{x} \).

2. **Condition for \( \lfloor x \rfloor \)**:
- For \( \lfloor x \rfloor > 0 \) and \( \lfloor x \rfloor \neq 1 \), we need \( x \geq 2 \) (since the greatest integer function will yield integers greater than or equal to 2).

3. **Condition for \( \frac{|x|}{x} \)**:
- If \( x > 0 \), then \( \frac{|x|}{x} = 1 \).
- If \( x < 0 \), then \( \frac{|x|}{x} = -1 \) (which is not valid for logarithm since logarithm of a negative number is undefined).

Thus, \( x \) must be greater than 0.

4. **Combining conditions**:
- From the above, we conclude that \( x \) must be in the interval \( [2, \infty) \).

### Step 2: Evaluate the Function

1. **Evaluate \( \frac{|x|}{x} \)**:
- For \( x \geq 2 \), \( \frac{|x|}{x} = 1 \).

2. **Substituting into the logarithm**:
- We have \( \log_{\lfloor x \rfloor}(1) = 0 \) for any \( \lfloor x \rfloor \) (since \( \log_a(1) = 0 \) for any \( a > 0 \)).

3. **Substituting into the function**:
- Therefore, \( f(x) = \cos^{-1}\left(\sqrt{0}\right) = \cos^{-1}(0) \).

4. **Finding the value of \( \cos^{-1}(0) \)**:
- The value of \( \cos^{-1}(0) \) is \( \frac{\pi}{2} \).

### Conclusion

The range of the function \( f(x) \) is a single value:

\[ \text{Range of } f(x) = \left\{ \frac{\pi}{2} \right\} \]