If `f: RvecR` is a function satisfying the property `f(2x+3)+f(2x+7)=2AAx in R ,` then find the fundamental period of `f(x)dot`

Explanation

To solve the problem step by step, we will analyze the functional equation given and derive the fundamental period of the function \( f(x) \).

### Step 1: Write down the given functional equation
The problem states that:
\[ f(2x + 3) + f(2x + 7) = 2 \quad \text{for all } x \in \mathbb{R} \]

### Step 2: Substitute \( x \) with \( x + 2 \)
To explore the properties of the function, we substitute \( x \) with \( x + 2 \):
\[ f(2(x + 2) + 3) + f(2(x + 2) + 7) = 2 \]
This simplifies to:
\[ f(2x + 4 + 3) + f(2x + 4 + 7) = 2 \]
or
\[ f(2x + 7) + f(2x + 11) = 2 \]

### Step 3: Label the equations
Let’s label the equations we have:
- Equation (1): \( f(2x + 3) + f(2x + 7) = 2 \)
- Equation (2): \( f(2x + 7) + f(2x + 11) = 2 \)

### Step 4: Subtract Equation (1) from Equation (2)
Now we will subtract Equation (1) from Equation (2):
\[ (f(2x + 7) + f(2x + 11)) - (f(2x + 3) + f(2x + 7)) = 0 \]
This simplifies to:
\[ f(2x + 11) - f(2x + 3) = 0 \]
Thus, we have:
\[ f(2x + 11) = f(2x + 3) \]

### Step 5: Relate the two expressions
From the equation \( f(2x + 11) = f(2x + 3) \), we can express this as:
\[ f(2x + 3) = f(2x + 3 + 8) \]
This means:
\[ f(t) = f(t + 8) \quad \text{where } t = 2x + 3 \]

### Step 6: Conclude the fundamental period
Since we have shown that \( f(t) = f(t + 8) \) for all \( t \), we conclude that the function \( f(x) \) is periodic with a fundamental period of:
\[ \text{Fundamental period} = 8 \]

### Final Answer
The fundamental period of \( f(x) \) is \( 8 \).

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