Statement-1: Function `f(x)=sin(x+3 sin x)` is periodic .
Statement-2: If g(x) is periodic then f(g(x)) periodic

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 determine whether the statements are true or false, we need to analyze each statement step by step.

### Step 1: Analyze Statement 1
The function given is \( f(x) = \sin(x + 3 \sin x) \).

1. **Understanding Periodicity**: A function \( f(x) \) is periodic if there exists a positive number \( T \) such that \( f(x + T) = f(x) \) for all \( x \).

2. **Check the inner function**: The term \( 3 \sin x \) is periodic with a period of \( 2\pi \). The sine function itself is periodic with a period of \( 2\pi \).

3. **Combine the terms**: The expression \( x + 3 \sin x \) does not have a fixed period because as \( x \) increases, the term \( x \) grows without bound, while \( 3 \sin x \) oscillates between -3 and 3.

4. **Conclusion for Statement 1**: Since the term \( x \) grows indefinitely, \( f(x) = \sin(x + 3 \sin x) \) does not repeat its values over any finite interval. Therefore, **Statement 1 is false**.

### Step 2: Analyze Statement 2
Statement 2 claims that if \( g(x) \) is periodic, then \( f(g(x)) \) is also periodic.

1. **Understanding the implication**: If \( g(x) \) is periodic with period \( T \), then \( g(x + T) = g(x) \) for all \( x \).

2. **Apply periodicity to \( f(g(x)) \)**: We need to check if \( f(g(x + T)) = f(g(x)) \).

3. **Substituting the periodicity of \( g(x) \)**:
\[ f(g(x + T)) = f(g(x)) \quad \text{(since \( g(x + T) = g(x) \))} \]

4. **Conclusion for Statement 2**: Since \( f(g(x + T)) = f(g(x)) \), it follows that \( f(g(x)) \) is periodic with the same period \( T \). Therefore, **Statement 2 is true**.

### Final Conclusion
- **Statement 1**: False (the function \( f(x) = \sin(x + 3 \sin x) \) is not periodic).
- **Statement 2**: True (if \( g(x) \) is periodic, then \( f(g(x)) \) is also periodic).