If f is an odd function and `f(a)=b`, which of the following must also be true ?
I. `f(a)=-b`
II.`f(-a)=b`
III. `f(-a)=-b`

Options

A. only I
B. Only III
C. Only III
D. Only I and II

Explanation

To solve the problem, we need to analyze the properties of an odd function and the implications of the given information.

### Step-by-Step Solution:

1. **Understanding Odd Functions**:
An odd function \( f(x) \) satisfies the property:
\[ f(-x) = -f(x) \]
This means that if you take the function at the negative of any input, the output will be the negative of the function at that input.

2. **Given Information**:
We are given that \( f(a) = b \).

3. **Applying the Property of Odd Functions**:
To find \( f(-a) \), we apply the property of odd functions:
\[ f(-a) = -f(a) \]
Since we know \( f(a) = b \), we substitute this into the equation:
\[ f(-a) = -b \]

4. **Evaluating the Statements**:
- **Statement I**: \( f(a) = -b \)
- This is false because we know \( f(a) = b \).
- **Statement II**: \( f(-a) = b \)
- This is also false because we found \( f(-a) = -b \).
- **Statement III**: \( f(-a) = -b \)
- This is true as derived from the property of odd functions.

5. **Conclusion**:
The only statement that must be true is Statement III: \( f(-a) = -b \).

### Final Answer:
The correct answer is Statement III only.