If the graph of the function y = f(x) is symmetrical about the line x = 2, then

Options

A. `f(x+2)=f(x-2)`
B. `f(2+x)=f(2-x)`
C. `f(x)=f(-x)`
D. `f(x)=-f(-x)`

Explanation

To solve the problem, we need to analyze the condition given: the function \( y = f(x) \) is symmetrical about the line \( x = 2 \). This symmetry implies a specific relationship between the function values at points equidistant from the line \( x = 2 \).

### Step-by-Step Solution:

1. **Understanding Symmetry**:
The statement that the graph of \( y = f(x) \) is symmetrical about the line \( x = 2 \) means that for any point \( (x, f(x)) \) on the graph, there is a corresponding point \( (4 - x, f(x)) \) on the graph. This is because if you take a point \( x \) units to the right of \( x = 2 \) (which is \( 2 + x \)), there should be a corresponding point \( x \) units to the left of \( x = 2 \) (which is \( 2 - x \)) that has the same function value.

2. **Setting Up the Equation**:
From the symmetry, we can express this relationship mathematically:
\[ f(2 + x) = f(2 - x) \]
This equation states that the function value at \( 2 + x \) is equal to the function value at \( 2 - x \).

3. **Analyzing the Options**:
We need to evaluate the given options to find which one matches our derived equation \( f(2 + x) = f(2 - x) \).

- **Option 1**: \( f(x + 2) = x - 2 \) - This does not represent the symmetry about \( x = 2 \).
- **Option 2**: \( f(2 + x) = 2 - x \) - This matches our derived equation.
- **Option 3**: \( f(x) = f(-x) \) - This represents even symmetry about the y-axis, not about \( x = 2 \).
- **Option 4**: Not provided in the transcript, but we can assume it does not match the derived equation.

4. **Conclusion**:
The correct answer is that the function satisfies the equation:
\[ f(2 + x) = f(2 - x) \]
Therefore, the correct option is **Option 2**.