The graph of `f:R rarr R` defined by y=f(x) is symmetric with respect to x=a and x=b. Which of the following is true ?

Options

A. f(2a-x)=f(x)
B. f(2a+x)=f(-x)
C. f(2b+x)=f(-x)
D. f is periodic

Answer

To solve the problem, we need to analyze the symmetry of the function \( f(x) \) with respect to the lines \( x = a \) and \( x = b \).

### Step-by-Step Solution:

1. **Understanding Symmetry**:
- A function \( f(x) \) is symmetric with respect to a vertical line \( x = c \) if \( f(c + x) = f(c - x) \) for all \( x \). This means that for every point on one side of the line, there is a corresponding point on the other side that has the same function value.

2. **Symmetry with respect to \( x = a \)**:
- For symmetry with respect to \( x = a \), we have:
\[ f(a + x) = f(a - x) \]
- This is our first equation (let's call it Equation 1).

3. **Symmetry with respect to \( x = b \)**:
- Similarly, for symmetry with respect to \( x = b \), we have:
\[ f(b + x) = f(b - x) \]
- This is our second equation (let's call it Equation 2).

4. **Substituting in Equation 1**:
- We can substitute \( x \) with \( a - x \) in Equation 1:
\[ f(a + (a - x)) = f(a - (a - x)) \]
- This simplifies to:
\[ f(2a - x) = f(x) \]
- This means:
\[ f(x) = f(2a - x) \]

5. **Substituting in Equation 2**:
- Now, we substitute \( x \) with \( b - x \) in Equation 2:
\[ f(b + (b - x)) = f(b - (b - x)) \]
- This simplifies to:
\[ f(2b - x) = f(x) \]
- This means:
\[ f(x) = f(2b - x) \]

6. **Conclusion**:
- From the above steps, we have derived two important properties:
- \( f(x) = f(2a - x) \)
- \( f(x) = f(2b - x) \)
- This indicates that the function \( f(x) \) is periodic with respect to the points \( 2a \) and \( 2b \).

### Final Result:
The function \( f(x) \) is symmetric with respect to both \( x = a \) and \( x = b \), leading us to conclude that it is periodic.