If f (x) is a function which is both odd and even then f (3) -f (2) is equal to

Options

A. 1
B. `-1`
C. `0`
D. None

Answer

To solve the problem, we need to analyze the properties of the function \( f(x) \) given that it is both odd and even.

### Step-by-Step Solution:

1. **Understanding Odd and Even Functions:**
- A function \( f(x) \) is **even** if \( f(-x) = f(x) \) for all \( x \).
- A function \( f(x) \) is **odd** if \( f(-x) = -f(x) \) for all \( x \).

2. **Analyzing the Given Condition:**
- The problem states that \( f(x) \) is both odd and even. This means:
- From the even condition: \( f(-x) = f(x) \)
- From the odd condition: \( f(-x) = -f(x) \)

3. **Setting the Equations Equal:**
- Since both conditions must hold true, we can set them equal to each other:
\[ f(x) = -f(x) \]

4. **Solving the Equation:**
- Rearranging the equation gives:
\[ 2f(x) = 0 \]
- Therefore, we find:
\[ f(x) = 0 \]
- This means that the function \( f(x) \) is the zero function, which is constant and equal to zero for all \( x \).

5. **Calculating \( f(3) - f(2) \):**
- Now, we can compute \( f(3) - f(2) \):
\[ f(3) = 0 \quad \text{and} \quad f(2) = 0 \]
- Thus:
\[ f(3) - f(2) = 0 - 0 = 0 \]

### Final Answer:
\[ f(3) - f(2) = 0 \]