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. none of these

Answer

To determine the properties of the function \( y = f(x) \) that is symmetric about the line \( x = 2 \), we can follow these steps:

### Step 1: Understand Symmetry about a Vertical Line
A function is symmetric about a vertical line \( x = a \) if for every point \( (a + d, f(a + d)) \) on the graph, there is a corresponding point \( (a - d, f(a - d)) \). In this case, since the line of symmetry is \( x = 2 \), we can say that:

\[ f(2 + d) = f(2 - d) \]

for any value of \( d \).

### Step 2: Substitute \( d \) with \( x - 2 \)
To express this symmetry in terms of \( x \), we can set \( d = x - 2 \). This leads to:

\[ f(2 + (x - 2)) = f(2 - (x - 2)) \]

which simplifies to:

\[ f(x) = f(4 - x) \]

This equation indicates that the function \( f(x) \) is symmetric about the line \( x = 2 \).

### Step 3: Conclusion
Thus, the final result we derive from the symmetry condition is:

\[ f(x) = f(4 - x) \]

This means that for any \( x \), the value of the function at \( x \) is equal to the value of the function at \( 4 - x \).

### Summary
The function \( y = f(x) \) is symmetric about the line \( x = 2 \) if and only if:

\[ f(x) = f(4 - x) \]