Let `f:R rarr R` defined by `f(x)=cos^(-1)(-{-x}),` where {x} denotes fractional part of x. Then, which of the following is/are correct?

Options

A. f is many one but not even function
B. Range of f contains two prime numbers
C. f is non-periodic
D. Graphs of f does not lie below X-axis

Answer

To solve the problem, we need to analyze the function \( f(x) = \cos^{-1}(-\{-x\}) \), where \(\{x\}\) denotes the fractional part of \(x\). The fractional part of \(x\) is defined as \(\{x\} = x - \lfloor x \rfloor\), where \(\lfloor x \rfloor\) is the greatest integer less than or equal to \(x\).

### Step-by-Step Solution:

1. **Understanding the Fractional Part**:
The fractional part \(\{-x\}\) can be expressed as:
\[ \{-x\} = -x - \lfloor -x \rfloor \]
Since \(\lfloor -x \rfloor = -\lfloor x \rfloor - 1\) if \(x\) is not an integer, we can rewrite it as:
\[ \{-x\} = -x + \lfloor x \rfloor + 1 \]
Thus, the fractional part \(\{-x\}\) is always in the range \([0, 1)\).

2. **Substituting into the Function**:
Now substituting this into \(f(x)\):
\[ f(x) = \cos^{-1}(-\{-x\}) = \cos^{-1}(-(-x + \lfloor x \rfloor + 1)) \]
Simplifying further, we have:
\[ f(x) = \cos^{-1}(x - \lfloor x \rfloor - 1) \]

3. **Analyzing the Range of \(f(x)\)**:
The term \(-\{-x\}\) will also be in the range \([-1, 0)\) as \(\{-x\} \in [0, 1)\). Therefore, the input to the \(\cos^{-1}\) function will be in the range \([-1, 0)\).

4. **Determining the Range of \(f(x)\)**:
The range of \(\cos^{-1}(y)\) for \(y \in [-1, 0)\) is:
\[ [\pi, \frac{\pi}{2}) \]
Therefore, the range of \(f(x)\) is \([\frac{\pi}{2}, \pi]\).

5. **Checking if \(f(x)\) is Many-One or Even**:
- **Many-One**: A function is many-one if different inputs can produce the same output. Given the periodic nature of the cosine function, \(f(x)\) can take the same value for different \(x\) values.
- **Even Function**: A function \(f\) is even if \(f(-x) = f(x)\). We can check this by substituting \(-x\) into the function and comparing it with \(f(x)\).

6. **Conclusion**:
Based on the analysis:
- The function is many-one.
- The function is not even.
- The range is \([\frac{\pi}{2}, \pi]\).

### Final Answer:
The correct options are:
- The function \(f(x)\) is many-one.
- The function \(f(x)\) is not an even function.