If `f(-x)=-f(x)` , then f(x) is

Options

A. an even function
B. an odd function
C. neither odd nor even
D. periodic function

Answer

To solve the problem, we need to analyze the given condition: \( f(-x) = -f(x) \).

### Step-by-Step Solution:

1. **Understanding the Property**: The equation \( f(-x) = -f(x) \) indicates a specific relationship between the function values at \( x \) and \( -x \). This property is characteristic of odd functions.

2. **Definition of Odd Functions**: A function \( f(x) \) is defined as an odd function if for every \( x \) in its domain, the following holds true:
\[ f(-x) = -f(x) \]
This means that the graph of the function is symmetric about the origin.

3. **Verifying the Given Condition**: Since the condition provided in the question matches the definition of an odd function, we can conclude that \( f(x) \) must be an odd function.

4. **Conclusion**: Therefore, the answer to the question is that \( f(x) \) is an odd function.

### Final Answer:
\( f(x) \) is an odd function.

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