Find the period `f(x)=sinx+{x},` where {x} is the fractional part of `xdot`

Answer

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

### Step-by-Step Solution:

1. **Identify the components of the function**:
The function \( f(x) \) consists of two parts:
- \( \sin x \)
- \( \{x\} \) (the fractional part of \( x \))

2. **Determine the period of \( \sin x \)**:
The sine function \( \sin x \) has a well-known period of \( 2\pi \). This means:
\[ \sin(x + 2\pi) = \sin x \]

3. **Determine the period of \( \{x\} \)**:
The fractional part function \( \{x\} \) is defined as:
\[ \{x\} = x - \lfloor x \rfloor \]
The fractional part function has a period of \( 1 \). This means:
\[ \{x + 1\} = \{x\} \]

4. **Finding the period of \( f(x) = \sin x + \{x\} \)**:
To find the overall period of the function \( f(x) \), we need to find the least common multiple (LCM) of the periods of the two components:
- Period of \( \sin x \) is \( 2\pi \)
- Period of \( \{x\} \) is \( 1 \)

5. **Calculate the LCM of \( 2\pi \) and \( 1 \)**:
Since \( 2\pi \) is an irrational number and \( 1 \) is a rational number, the LCM of an irrational number and a rational number does not exist in the traditional sense. Therefore, we conclude that:
\[ \text{LCM}(2\pi, 1) \text{ does not exist.} \]

6. **Conclusion about the periodicity of \( f(x) \)**:
Since the LCM does not exist, the function \( f(x) = \sin x + \{x\} \) is non-periodic.

### Final Answer:
The function \( f(x) = \sin x + \{x\} \) is non-periodic.