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 it satisfies the condition:
\[ f(a - d) = f(a + d) \]
for all values of \( d \). This means that the function's values at equal distances from the line of symmetry are the same.

### Step 2: Applying Symmetry Conditions
Given that the function is symmetric about both \( x = 2 \) and \( x = 3 \), we can set up the following equations based on the symmetry conditions:

1. For symmetry about \( x = 2 \):
\[ f(2 - d) = f(2 + d) \]

2. For symmetry about \( x = 3 \):
\[ f(3 - d) = f(3 + d) \]

### Step 3: Finding a General Form
To find a function that satisfies both conditions, we can express the function in terms of periodic functions. A periodic function has the property that it repeats its values at regular intervals.

### Step 4: Periodicity
If we consider a periodic function with a period that accommodates both lines of symmetry, we can use a function such as:
\[ f(x) = f(2 + k) \text{ and } f(x) = f(3 + k) \]
where \( k \) is a constant that defines the periodicity.

### Step 5: Example Functions
A simple example of such a function could be:
\[ f(x) = \sin(\pi (x - 2)) \]
This function is periodic with a period of 2, which means it will repeat its values around both lines of symmetry.

### Conclusion
Thus, the functions that are symmetric about \( x = 2 \) and \( x = 3 \) can be represented as periodic functions, such as:
\[ f(x) = \sin(\pi (x - 2)) \]
or any other function that satisfies the periodicity condition.