Statement -1: f(x) is invertible function and f(f(x)) = `x AA x in` domain of f(x), then f(x) is symmetrical about the line y = x.
Statement -2: f(x) and `f^(-1)(x)` are symmetrical about the line y = x.

Options

A. Statement -1 is True, Statement -2 is True, Statement-2 is a correct explanation for Statement-1
B. Statement -1 is True, Statement -2 is True,. 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 problem, we need to analyze the two statements provided and determine their validity step by step.

### Step 1: Understanding Statement 1
**Statement 1:** If \( f(x) \) is an invertible function and \( f(f(x)) = x \) for all \( x \) in the domain of \( f(x) \), then \( f(x) \) is symmetrical about the line \( y = x \).

- An invertible function \( f(x) \) means that there exists a function \( f^{-1}(x) \) such that \( f(f^{-1}(x)) = x \) for all \( x \) in the range of \( f \).
- The condition \( f(f(x)) = x \) implies that applying the function \( f \) twice returns the original input \( x \). This suggests that \( f \) is its own inverse, i.e., \( f^{-1}(x) = f(x) \).
- For a function to be symmetrical about the line \( y = x \), it must satisfy the property that if \( (a, b) \) is a point on the graph of the function, then \( (b, a) \) must also be a point on the graph.
- Since \( f(f(x)) = x \), we can deduce that \( f(x) \) must indeed reflect across the line \( y = x \).

**Conclusion for Statement 1:** True.

### Step 2: Understanding Statement 2
**Statement 2:** \( f(x) \) and \( f^{-1}(x) \) are symmetrical about the line \( y = x \).

- By definition, the inverse function \( f^{-1}(x) \) is obtained by reflecting the graph of \( f(x) \) across the line \( y = x \).
- Therefore, if \( f(x) \) is symmetrical about the line \( y = x \), then \( f^{-1}(x) \) must also be symmetrical about the same line.
- Since we established from Statement 1 that \( f(x) \) is symmetrical about \( y = x \), it follows that \( f^{-1}(x) \) is also symmetrical about \( y = x \).

**Conclusion for Statement 2:** True.

### Step 3: Relationship Between the Statements
- Statement 2 serves as a correct explanation for Statement 1. Since both statements are true and Statement 2 explains why Statement 1 is true, we can conclude that the overall assertion is valid.

### Final Conclusion
Both statements are true, and Statement 2 is the correct explanation for Statement 1.

### Summary of the Solution
1. **Statement 1:** True (f(x) is symmetrical about y = x).
2. **Statement 2:** True (f(x) and f^(-1)(x) are symmetrical about y = x).
3. **Relationship:** Statement 2 explains Statement 1.