Which of the following fucntion has period 2?

Options

A. `cos[((pi)/(4))x]`
B. `cos[((pi)/(2))x]`
C. `cos(2pix)`
D. `cos(pix)`

Explanation

To determine which of the given functions has a period of 2, we will calculate the period of each function step by step.

### Step 1: Understand the Period of Cosine Function
The standard period of the cosine function \( \cos(x) \) is \( 2\pi \). When the argument of the cosine function is modified, the period changes according to the formula:
\[ \text{Period} = \frac{2\pi}{k} \]
where \( k \) is the coefficient of \( x \) in the argument of the cosine function.

### Step 2: Calculate the Period of Each Function

1. **Function: \( \cos\left(\frac{\pi}{4} x\right) \)**
- Here, \( k = \frac{\pi}{4} \).
- Period = \( \frac{2\pi}{\frac{\pi}{4}} = 2\pi \cdot \frac{4}{\pi} = 8 \).
- **Conclusion**: This function does not have a period of 2.

2. **Function: \( \cos\left(\frac{\pi}{2} x\right) \)**
- Here, \( k = \frac{\pi}{2} \).
- Period = \( \frac{2\pi}{\frac{\pi}{2}} = 2\pi \cdot \frac{2}{\pi} = 4 \).
- **Conclusion**: This function does not have a period of 2.

3. **Function: \( \cos(2\pi x) \)**
- Here, \( k = 2\pi \).
- Period = \( \frac{2\pi}{2\pi} = 1 \).
- **Conclusion**: This function does not have a period of 2.

4. **Function: \( \cos(\pi x) \)**
- Here, \( k = \pi \).
- Period = \( \frac{2\pi}{\pi} = 2 \).
- **Conclusion**: This function has a period of 2.

### Final Answer
The function that has a period of 2 is \( \cos(\pi x) \).

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