IF a function is symmetric about the x =2 and x = 3, then the functions are

Options

A. all periodic functions with fundametnal period 3
B. all periodic functions with fundametnal period 2
C. all periodic functions with fundametnal period 1
D. none of these

Answer

To determine the functions that are symmetric about the lines \(x = 2\) and \(x = 3\), we can follow these steps:

### Step 1: Understanding Symmetry
A function \(f(x)\) is symmetric about a vertical line \(x = a\) if:
\[ f(a - x) = f(a + x) \]
This means that the function behaves the same way on both sides of the line \(x = a\).

### Step 2: Applying Symmetry about \(x = 2\)
For symmetry about \(x = 2\):
\[ f(2 - x) = f(2 + x) \]
This implies that if we replace \(x\) with \(2 - x\) and \(2 + x\), the function values should be equal.

### Step 3: Applying Symmetry about \(x = 3\)
For symmetry about \(x = 3\):
\[ f(3 - x) = f(3 + x) \]
Similarly, this means that the function values are equal when we replace \(x\) with \(3 - x\) and \(3 + x\).

### Step 4: Combining the Conditions
From the above two conditions, we can derive that:
1. From \(x = 2\): \(f(2 - x) = f(2 + x)\)
2. From \(x = 3\): \(f(3 - x) = f(3 + x)\)

### Step 5: Finding a General Form
To satisfy both conditions, we can express \(f(x)\) in terms of periodicity. If we take \(f(x) = f(x + 2)\), it indicates that the function is periodic with a period of 2.

### Step 6: Conclusion
Thus, the functions that are symmetric about both \(x = 2\) and \(x = 3\) are periodic functions with a period of 2.

### Final Answer
The functions are periodic with a period of 2.

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