Find period of following functions (if existsI)
`f(x)=x-[x]+cos(pix)`

Answer

To find the period of the function \( f(x) = x - [x] + \cos(\pi x) \), we will break it down into its components and analyze their periodicity.

### Step-by-Step Solution:

1. **Identify Components of the Function**:
The function can be expressed as:
\[ f(x) = (x - [x]) + \cos(\pi x) \]
Here, \( [x] \) is the greatest integer function (also known as the floor function), and \( x - [x] \) represents the fractional part of \( x \).

2. **Analyze the Fractional Part**:
The fractional part of \( x \), denoted as \( \{x\} = x - [x] \), is periodic with a period of 1. This is because:
\[ \{x + 1\} = (x + 1) - [x + 1] = (x + 1) - ([x] + 1) = x - [x] = \{x\} \]

3. **Analyze the Cosine Component**:
The function \( \cos(\pi x) \) is a cosine function which is periodic. The period of \( \cos(kx) \) is given by:
\[ \text{Period} = \frac{2\pi}{k} \]
In our case, \( k = \pi \), so the period of \( \cos(\pi x) \) is:
\[ \text{Period} = \frac{2\pi}{\pi} = 2 \]

4. **Determine the Overall Period**:
Since \( f(x) \) is the sum of two periodic functions, we need to find the least common multiple (LCM) of their periods to determine the period of \( f(x) \).
- The period of \( \{x\} \) is 1.
- The period of \( \cos(\pi x) \) is 2.

The LCM of 1 and 2 is:
\[ \text{LCM}(1, 2) = 2 \]

5. **Conclusion**:
Therefore, the period of the function \( f(x) = x - [x] + \cos(\pi x) \) is:
\[ \boxed{2} \]