Let `f : R rarr[-1,oo] and f(x)= ln([|sin 2 x|+|cos 2 x|])` (where[.] is greatest integer function), then -

Options

A. `f(x)` has range Z
B. Range of `f(x)` is singleton set
C. `f(x)` is invertible in `[0,(pi)/(4)]`
D. `f(x)` is into function

Explanation

To solve the problem, we need to analyze the function \( f(x) = \ln(|\sin 2x| + |\cos 2x|) \) and determine its properties, particularly its range.

### Step 1: Analyze the function inside the logarithm
We start by examining the expression \( |\sin 2x| + |\cos 2x| \).

**Hint:** Recall the properties of sine and cosine functions, particularly their maximum and minimum values.

### Step 2: Determine the maximum and minimum values of \( |\sin 2x| + |\cos 2x| \)
The maximum value of \( |\sin 2x| + |\cos 2x| \) occurs when both sine and cosine are at their maximum values. We know that:
- The maximum value of \( |\sin 2x| \) is 1.
- The maximum value of \( |\cos 2x| \) is also 1.

Thus, the maximum value of \( |\sin 2x| + |\cos 2x| \) is:
\[ |\sin 2x| + |\cos 2x| \leq 1 + 1 = 2. \]

**Hint:** Consider specific angles where sine and cosine take these maximum values.

### Step 3: Find the minimum value of \( |\sin 2x| + |\cos 2x| \)
The minimum value occurs when both sine and cosine are zero, but since they cannot be zero simultaneously, we need to check other combinations.

The minimum value occurs at:
\[ |\sin 2x| + |\cos 2x| \geq 1. \]
This can be shown using the Cauchy-Schwarz inequality or by considering specific angles.

**Hint:** Think about the unit circle and the angles at which sine and cosine are equal.

### Step 4: Conclude the range of \( |\sin 2x| + |\cos 2x| \)
From the above analysis, we conclude:
\[ 1 \leq |\sin 2x| + |\cos 2x| \leq 2. \]

**Hint:** Remember that the logarithm function is defined only for positive values.

### Step 5: Apply the logarithm
Now, we apply the logarithm to the range:
\[ \ln(1) \leq \ln(|\sin 2x| + |\cos 2x|) \leq \ln(2). \]
This simplifies to:
\[ 0 \leq f(x) \leq \ln(2). \]

**Hint:** The greatest integer function will affect the final range.

### Step 6: Apply the greatest integer function
Since \( f(x) \) can take values from \( 0 \) to \( \ln(2) \), we need to find the greatest integer less than or equal to \( f(x) \).

Given that \( \ln(2) \approx 0.693 \), we can conclude:
\[ f(x) \in [0, \ln(2)] \implies [f(x)] \in [0, 0]. \]

**Hint:** Consider the implications of the greatest integer function on the range.

### Final Conclusion
Thus, the range of the function \( f(x) = \ln(|\sin 2x| + |\cos 2x|) \) when applying the greatest integer function is:
\[ f(x) \in \{0\}. \]

This means that the function is constant at 0 for all \( x \) in \( \mathbb{R} \).

### Summary of Steps:
1. Analyze the function inside the logarithm.
2. Determine the maximum and minimum values of \( |\sin 2x| + |\cos 2x| \).
3. Conclude the range of \( |\sin 2x| + |\cos 2x| \).
4. Apply the logarithm to find the range of \( f(x) \).
5. Apply the greatest integer function to find the final range.