Statement-1 : f(x) is a one-one function `hArr f^(-1) (x)` is a one-one function.
and
Statement-2 `f^(-1)(x)` is the reflection of the function f(x) with respect to 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 False, Statement -2 is False
D. Statement -1 is False, Statement -2 is True

Answer

To solve the question, we need to analyze both statements provided and establish their validity.

### Step 1: Understand Statement 1
**Statement 1:** \( f(x) \) is a one-one function \( \Rightarrow f^{-1}(x) \) is a one-one function.

- A function \( f(x) \) is said to be one-one (or injective) if it assigns distinct outputs to distinct inputs. This means that for any two different values \( a \) and \( b \) in the domain, \( f(a) \neq f(b) \).
- If \( f(x) \) is one-one, it implies that every output \( y \) in the range corresponds to exactly one input \( x \). Therefore, the inverse function \( f^{-1}(x) \) will also be one-one because it will also assign distinct outputs to distinct inputs.

**Conclusion for Statement 1:** True.

### Step 2: Understand Statement 2
**Statement 2:** \( f^{-1}(x) \) is the reflection of the function \( f(x) \) with respect to the line \( y = x \).

- The graph of the inverse function \( f^{-1}(x) \) can be obtained by reflecting the graph of \( f(x) \) across the line \( y = x \). This means that if the point \( (a, b) \) lies on the graph of \( f(x) \), then the point \( (b, a) \) will lie on the graph of \( f^{-1}(x) \).
- This property holds true for all functions and their inverses, reinforcing that the inverse function is indeed a reflection of the original function across the line \( y = x \).

**Conclusion for Statement 2:** True.

### Step 3: Relate the Statements
Since both statements are true, we can conclude that Statement 2 serves as a valid explanation for Statement 1. The reflection property of functions and their inverses confirms that if \( f(x) \) is one-one, then \( f^{-1}(x) \) is also one-one.

### Final Conclusion
Both statements are correct, and Statement 2 provides a valid explanation for Statement 1.