Period of the function `f(x)=sin(sin(pix))+e^({3x})`, where {.} denotes the fractional part of x is

Options

A. 1
B. 2
C. 3
D. None of these

Answer

To find the period of the function \( f(x) = \sin(\sin(\pi x)) + e^{3x} \), we will analyze the periods of the individual components of the function.

### Step 1: Identify the components of the function
The function \( f(x) \) consists of two parts:
1. \( g_1(x) = \sin(\sin(\pi x)) \)
2. \( g_2(x) = e^{3x} \)

### Step 2: Determine the period of \( g_1(x) = \sin(\sin(\pi x)) \)
The inner function \( \sin(\pi x) \) has a period of \( 2 \) because:
- The function \( \sin(x) \) has a period of \( 2\pi \).
- Thus, \( \sin(\pi x) \) has a period of \( \frac{2\pi}{\pi} = 2 \).

Now, since \( \sin(\sin(\pi x)) \) is a composition of periodic functions, it retains the same period. Therefore, the period of \( g_1(x) \) is \( 2 \).

### Step 3: Determine the period of \( g_2(x) = e^{3x} \)
The function \( e^{3x} \) is an exponential function and does not have a finite period. However, we can consider the fractional part of \( 3x \) denoted by \( \{3x\} \), which has a period of \( \frac{1}{3} \) because:
- The fractional part function \( \{x\} \) has a period of \( 1 \).
- Therefore, \( \{3x\} \) has a period of \( \frac{1}{3} \).

### Step 4: Find the overall period of \( f(x) \)
To find the overall period of \( f(x) \), we need to find the least common multiple (LCM) of the periods of \( g_1(x) \) and \( g_2(x) \):
- The period of \( g_1(x) \) is \( 2 \).
- The period of \( g_2(x) \) is \( \frac{1}{3} \).

To find the LCM of \( 2 \) and \( \frac{1}{3} \):
- Convert \( 2 \) into a fraction: \( 2 = \frac{6}{3} \).
- The LCM of \( \frac{6}{3} \) and \( \frac{1}{3} \) is \( \frac{6}{3} = 2 \).

### Conclusion
The period of the function \( f(x) = \sin(\sin(\pi x)) + e^{3x} \) is \( 2 \).