The function `f(x) = 3^(sin^(2)pi + sin^(4) pi x + x-[x])` where [x] denotes the greatest interger less than or equal to x, is

Options

A. a periodic function with period 1
B. a periodic function with period 2
C. a periodic function with period ` (1)/(2)`
D. not a periodic function

Explanation

To determine whether the function \( f(x) = 3^{\sin^2(\pi) + \sin^4(\pi x) + x - [x]} \) is periodic, we will analyze the components of the function step by step.

### Step 1: Simplify the Function
First, we note that \( \sin^2(\pi) = 0 \). Therefore, we can simplify the function:

\[ f(x) = 3^{0 + \sin^4(\pi x) + x - [x]} = 3^{\sin^4(\pi x) + x - [x]} \]

### Step 2: Analyze \( \sin^4(\pi x) \)
The term \( \sin^4(\pi x) \) is periodic. The sine function, \( \sin(\pi x) \), has a period of 2, which means \( \sin^4(\pi x) \) also has a period of 2.

### Step 3: Analyze \( x - [x] \)
The term \( x - [x] \) represents the fractional part of \( x \), denoted as \( \{x\} \). The fractional part function is periodic with a period of 1, since it repeats every integer value.

### Step 4: Combine the Periods
Now we have two components:
1. \( \sin^4(\pi x) \) has a period of 2.
2. \( x - [x] \) has a period of 1.

To find the overall period of the function \( f(x) \), we need to find the least common multiple (LCM) of the two periods.

- The LCM of 2 and 1 is 2.

### Step 5: Determine the Periodicity of \( f(x) \)
Since \( f(x) = 3^{\sin^4(\pi x) + x - [x]} \) is composed of periodic functions, we conclude that \( f(x) \) is periodic with a period of 2.

### Final Conclusion
Thus, the function \( f(x) \) is periodic with period 2.