The period of `f(x)=cosx+{x}` , is

Options

A. `2pi`
B. ` 1`
C. `pi`
D. none- existent

Explanation

To find the period of the function \( f(x) = \cos x + \{x\} \), where \(\{x\}\) represents the fractional part of \(x\), we will analyze the periods of the individual components of the function.

### Step 1: Identify the periods of the components
- The function \( \cos x \) has a period of \( 2\pi \).
- The fractional part function \( \{x\} = x - \lfloor x \rfloor \) has a period of \( 1 \).

### Step 2: Determine the combined period
To find the period of the combined function \( f(x) = \cos x + \{x\} \), we need to consider the least common multiple (LCM) of the two periods:
- Period of \( \cos x \) = \( 2\pi \)
- Period of \( \{x\} \) = \( 1 \)

### Step 3: Calculate the LCM
The LCM of \( 2\pi \) and \( 1 \):
- Since \( 2\pi \) is an irrational number and \( 1 \) is a rational number, the LCM does not exist in the conventional sense.

### Step 4: Conclusion
Since the LCM does not exist, the function \( f(x) = \cos x + \{x\} \) does not have a period. Therefore, we conclude that:

\[ \text{The period of } f(x) \text{ is non-existent.} \]

### Final Answer
The period of \( f(x) = \cos x + \{x\} \) is non-existent.

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