If a real function f(x) satisfies the relation `f(x) = f(2a -x)` for all `x in R`. Then, its graph is symmetrical about the line.

Options

A. ` x = 0`
B. ` x = 2a `
C. ` x = a `
D. ` x = -a `

Explanation

To solve the problem, we need to analyze the given functional equation and determine the symmetry of the function's graph.

### Step-by-Step Solution:

1. **Understanding the Given Relation**:
We start with the relation given in the problem:
\[ f(x) = f(2a - x) \quad \text{for all } x \in \mathbb{R} \]
This indicates that the function \( f(x) \) has a certain symmetry.

2. **Substituting \( x \) with \( a + x \)**:
To explore the symmetry further, we substitute \( x \) with \( a + x \):
\[ f(a + x) = f(2a - (a + x)) = f(a - x) \]
This step helps us to see how the function behaves when we shift \( x \) by \( a \).

3. **Identifying the Symmetry**:
From the equation \( f(a + x) = f(a - x) \), we can interpret this as follows:
- The function values at \( a + x \) and \( a - x \) are equal.
- This is the definition of symmetry about the vertical line \( x = a \).

4. **Conclusion**:
Since we have shown that \( f(a + x) = f(a - x) \), we conclude that the graph of the function \( f(x) \) is symmetric about the line \( x = a \).

### Final Answer:
The graph of the function \( f(x) \) is symmetric about the line \( x = a \).

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