If the graph of a function f(x) is symmetrical about the line x = a, then

Options

A. ` f (a + x ) = f (a- x )`
B. `f (a + x) = f (x - a )`
C. `f(x) = f (-x)`
D. none of these

Explanation

To determine the properties of a function \( f(x) \) that is symmetrical about the line \( x = a \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Symmetry**:
A function \( f(x) \) is said to be symmetrical about the line \( x = a \) if for every point \( (x, f(x)) \) on the graph, there exists a corresponding point \( (2a - x, f(x)) \). This means that the distance from \( x \) to \( a \) is the same as the distance from \( 2a - x \) to \( a \).

2. **Setting Up the Points**:
Let’s take a point \( (x, f(x)) \) on the graph of the function. If the graph is symmetrical about the line \( x = a \), then the corresponding point on the other side of the line would be \( (2a - x, f(x)) \).

3. **Using the Property of Symmetry**:
Since the y-coordinates of these two points must be equal due to symmetry, we can write:
\[ f(x) = f(2a - x) \]

4. **Conclusion**:
The equation \( f(x) = f(2a - x) \) indicates that the function \( f(x) \) is even with respect to the line \( x = a \). This means that the function takes the same value at points equidistant from the line \( x = a \).

### Final Result:
Thus, if the graph of a function \( f(x) \) is symmetrical about the line \( x = a \), we conclude that:
\[ f(x) = f(2a - x) \]