Which of the following is/are periodic?

Answer

To determine which of the given functions are periodic, we will analyze each function step by step.

### Step 1: Analyze the first function \( f(x) = x - \lfloor x \rfloor \)

1. **Understanding the function**: The function \( f(x) = x - \lfloor x \rfloor \) represents the fractional part of \( x \). It takes the value of \( x \) minus the greatest integer less than or equal to \( x \).

2. **Behavior in intervals**: This function behaves differently in each interval of integers. For example:
- In the interval \( [0, 1) \), \( f(x) = x - 0 = x \).
- In the interval \( [1, 2) \), \( f(x) = x - 1 \).
- In the interval \( [2, 3) \), \( f(x) = x - 2 \), and so on.

3. **Graphing**: The graph of \( f(x) \) will be a series of line segments that rise from 0 to 1 in each interval \( [n, n+1) \) where \( n \) is an integer, and it will reset to 0 at each integer.

4. **Periodicity**: Since the function repeats its values in every interval of length 1 (from \( n \) to \( n+1 \)), we conclude that \( f(x) \) is periodic with a period of 1.

### Step 2: Analyze the second function \( f(x) = x \)

1. **Understanding the function**: The function \( f(x) = x \) is a linear function.

2. **Graphing**: The graph of this function is a straight line that passes through the origin and continues indefinitely in both directions.

3. **Periodicity**: Since for every value of \( x \) there is a unique value of \( f(x) \), and it does not repeat any values, this function is not periodic.

### Step 3: Analyze the third function \( f(x) = 2 \tan\left(\frac{1}{2}x\right) - 5 \)

1. **Understanding the function**: The tangent function \( \tan(x) \) is periodic with a period of \( \pi \). The function \( \tan\left(\frac{1}{2}x\right) \) has a period of \( 2\pi \) because the argument is scaled by \( \frac{1}{2} \).

2. **Behavior of the function**: The function \( f(x) = 2 \tan\left(\frac{1}{2}x\right) - 5 \) will also be periodic with the same period as \( \tan\left(\frac{1}{2}x\right) \), which is \( 2\pi \).

3. **Graphing**: The graph will have vertical asymptotes where \( \tan\left(\frac{1}{2}x\right) \) is undefined, specifically at \( x = 2n\pi \) for integers \( n \).

4. **Periodicity**: Since the function repeats its values every \( 2\pi \), we conclude that this function is periodic with a period of \( 2\pi \).

### Conclusion

- The first function \( f(x) = x - \lfloor x \rfloor \) is periodic with a period of 1.
- The second function \( f(x) = x \) is not periodic.
- The third function \( f(x) = 2 \tan\left(\frac{1}{2}x\right) - 5 \) is periodic with a period of \( 2\pi \).

Thus, the periodic functions are the first and the third functions.