Statement-1 : If f(x) and g(x) are one-one functions then f(g(x)) and g(f(x)) is also a one-one function.
and
Statement-2 The composite function of two one-one function may or many not be one-one.

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 given statements regarding the one-one functions \( f(x) \) and \( g(x) \), we will analyze both statements step by step.

### Step 1: Understanding One-One Functions
A function \( f(x) \) is said to be one-one (or injective) if it satisfies the condition:
\[ f(x_1) = f(x_2) \implies x_1 = x_2 \]
This means that different inputs lead to different outputs.

### Step 2: Analyzing Statement 1
**Statement 1:** If \( f(x) \) and \( g(x) \) are one-one functions, then \( f(g(x)) \) and \( g(f(x)) \) are also one-one functions.

1. **Consider \( f(g(x_1)) = f(g(x_2)) \)**:
- Since \( f \) is one-one, we can conclude that:
\[ g(x_1) = g(x_2) \]

2. **Now, since \( g \) is also one-one**:
- From \( g(x_1) = g(x_2) \), we can conclude that:
\[ x_1 = x_2 \]

3. **Thus, we have shown that**:
\[ f(g(x_1)) = f(g(x_2)) \implies x_1 = x_2 \]
- Therefore, \( f(g(x)) \) is one-one.

4. **Now, consider \( g(f(x_1)) = g(f(x_2)) \)**:
- Similarly, since \( g \) is one-one, we have:
\[ f(x_1) = f(x_2) \]

5. **And since \( f \) is one-one**:
- From \( f(x_1) = f(x_2) \), we can conclude that:
\[ x_1 = x_2 \]

6. **Thus, we have shown that**:
\[ g(f(x_1)) = g(f(x_2)) \implies x_1 = x_2 \]
- Therefore, \( g(f(x)) \) is also one-one.

### Conclusion for Statement 1:
Since both \( f(g(x)) \) and \( g(f(x)) \) are shown to be one-one, **Statement 1 is true**.

### Step 3: Analyzing Statement 2
**Statement 2:** The composite function of two one-one functions may or may not be one-one.

- From our analysis in Statement 1, we have established that the composition of two one-one functions is always one-one.
- Therefore, the assertion that it "may or may not be one-one" is incorrect.

### Conclusion for Statement 2:
Since we have proven that the composition of two one-one functions is always one-one, **Statement 2 is false**.

### Final Answer:
- **Statement 1 is true.**
- **Statement 2 is false.**