If` f(x)` is an odd periodic function with period 2, then f(4) equals to-

Options

A. `-4`
B. 4
C. 2
D. 0

Explanation

To solve the problem, we need to find the value of \( f(4) \) given that \( f(x) \) is an odd periodic function with a period of 2.

### Step-by-Step Solution:

1. **Understanding the properties of the function**:
- Since \( f(x) \) is an odd function, it satisfies the property:
\[ f(-x) = -f(x) \]
- Since \( f(x) \) is periodic with a period of 2, it satisfies the property:
\[ f(x) = f(x + 2) \]

2. **Finding \( f(0) \)**:
- Using the odd function property, we can evaluate \( f(0) \):
\[ f(0) + f(0) = 0 \quad \text{(since } f(-0) = -f(0) \text{)} \]
- This simplifies to:
\[ 2f(0) = 0 \implies f(0) = 0 \]

3. **Using the periodic property**:
- Since \( f(x) \) is periodic with a period of 2, we can find \( f(2) \):
\[ f(2) = f(0) = 0 \]

4. **Finding \( f(4) \)**:
- Now, using the periodic property again:
\[ f(4) = f(4 - 2) = f(2) \]
- We already found that \( f(2) = 0 \), thus:
\[ f(4) = 0 \]

5. **Conclusion**:
- Therefore, the value of \( f(4) \) is:
\[ \boxed{0} \]