Statement 1 : f (x) = |x - 3| + |x - 4| + |x - 7| where `4 lt x lt 7` is an identity function.
Statement 2 : `f : A to A ` defined by f (x) = x is an identity function.

Options

A. Statement-1 is True, Statement-2 is True and Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True and Statement-2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To solve the question, we need to analyze the two statements provided and determine their validity regarding identity functions.

### Step-by-Step Solution:

**Step 1: Analyze Statement 1**

The first statement is:
\[ f(x) = |x - 3| + |x - 4| + |x - 7| \]
for \( 4 < x < 7 \).

To analyze this, we need to evaluate the function \( f(x) \) within the interval \( 4 < x < 7 \).

1. **Evaluate \( |x - 3| \)**:
- Since \( x > 4 \), \( x - 3 > 0 \).
- Therefore, \( |x - 3| = x - 3 \).

2. **Evaluate \( |x - 4| \)**:
- Since \( x > 4 \), \( x - 4 > 0 \).
- Therefore, \( |x - 4| = x - 4 \).

3. **Evaluate \( |x - 7| \)**:
- Since \( x < 7 \), \( x - 7 < 0 \).
- Therefore, \( |x - 7| = -(x - 7) = 7 - x \).

Now, substituting these values into \( f(x) \):
\[ f(x) = (x - 3) + (x - 4) + (7 - x) \]
\[ = x - 3 + x - 4 + 7 - x \]
\[ = x \]

Thus, within the interval \( 4 < x < 7 \), we have:
\[ f(x) = x \]

This confirms that \( f(x) \) acts as an identity function in the specified interval.

**Step 2: Analyze Statement 2**

The second statement is:
If \( f: A \to A \) is defined by \( f(x) = x \), then it is an identity function.

This statement is true because an identity function is defined as a function where every element in the domain maps to itself in the codomain. Therefore, \( f(x) = x \) is indeed an identity function.

**Step 3: Conclusion**

Both statements are true. However, Statement 2 does not explain Statement 1, as Statement 1 is specific to the function defined by absolute values in a certain interval while Statement 2 is a general definition of identity functions.

### Final Answer:
Both statements are true, but Statement 2 is not the correct explanation of Statement 1. Therefore, the correct option is:
**Option 2: Statement 1 and Statement 2 are both true, but Statement 2 is not the correct explanation of Statement 1.**