Consider that the graph of `y = f(x)` is symmetrie about the lines `x = 2 and x = 4` then the period of `f(x)` is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the period of the function \( f(x) \) that is symmetric about the lines \( x = 2 \) and \( x = 4 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Symmetry**:
- The function \( f(x) \) is symmetric about the lines \( x = 2 \) and \( x = 4 \). This means that for any point \( (a, f(a)) \) on the graph, there exists a corresponding point \( (4 - (a - 4), f(a)) \) and \( (2 - (a - 2), f(a)) \) that are also on the graph.

2. **Setting Up the Symmetry Conditions**:
- From the symmetry about \( x = 2 \), we have:
\[ f(2 - d) = f(2 + d) \]
- From the symmetry about \( x = 4 \), we have:
\[ f(4 - d) = f(4 + d) \]

3. **Expressing the Function**:
- Let’s denote \( a = 2 + d \) and \( b = 4 + d \). The symmetry conditions imply:
\[ f(2 - d) = f(a) \quad \text{and} \quad f(4 - d) = f(b) \]

4. **Finding the Period**:
- To find the period of \( f(x) \), we need to express \( f(x + T) = f(x) \) for some period \( T \).
- Since the function is symmetric about both lines, we can analyze the distance between these lines:
\[ \text{Distance between } x = 2 \text{ and } x = 4 = 4 - 2 = 2 \]
- Therefore, the function must repeat itself every distance of \( 4 \) (the distance from \( x = 2 \) to \( x = 4 \) and back).

5. **Conclusion**:
- The period \( T \) of the function \( f(x) \) is \( 4 \).

Thus, the period of \( f(x) \) is \( 4 \).