If[.] and {.} denote greatest integer and fractional part functions respectively, then the period of `f(x) = e^(sin 3pi{x} + tan pi [x])` is

Options

A. `2//3`
B. `1`
C. `3`
D. none of these

Explanation

To find the period of the function \( f(x) = e^{\sin(3\pi \{x\}) + \tan(\pi [x])} \), where \([x]\) denotes the greatest integer function and \(\{x\}\) denotes the fractional part function, we can analyze each component of the function separately.

### Step-by-step Solution:

1. **Understanding the Functions**:
- The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\).
- The fractional part function \(\{x\} = x - [x]\) gives the non-integer part of \(x\), which is periodic with a period of 1.

2. **Analyzing \(\tan(\pi [x])\)**:
- The function \(\tan(\pi [x])\) is periodic with a period of 1 because \([x]\) changes only when \(x\) increases by 1. Thus, \(\tan(\pi [x + 1]) = \tan(\pi [x] + \pi) = -\tan(\pi [x])\), which means that the function will repeat every integer interval.

3. **Analyzing \(\sin(3\pi \{x\})\)**:
- The fractional part function \(\{x\}\) is also periodic with a period of 1. Therefore, \(\sin(3\pi \{x + 1\}) = \sin(3\pi \{x\})\).

4. **Combining the Results**:
- Since both components of the function \( f(x) \) are periodic with a period of 1, we can conclude that \( f(x + 1) = e^{\sin(3\pi \{x + 1\}) + \tan(\pi [x + 1])} = e^{\sin(3\pi \{x\}) + \tan(\pi [x])} = f(x) \).

5. **Conclusion**:
- Therefore, the period of the function \( f(x) \) is 1.

### Final Answer:
The period of \( f(x) = e^{\sin(3\pi \{x\}) + \tan(\pi [x])} \) is \( 1 \).