The period of the function `f(x)=Sin(x +3-[x+3]) `where [] denotes the greatest integer function

Options

A. `2pi+3`
B. `2pi`
C. 1
D. 4

Answer

To find the period of the function \( f(x) = \sin(x + 3 - [x + 3]) \), where \( [ \cdot ] \) denotes the greatest integer function, we can follow these steps:

### Step 1: Simplify the function using the greatest integer function
The expression \( x + 3 - [x + 3] \) represents the fractional part of \( x + 3 \). We can denote the fractional part of a number \( y \) as \( \{y\} = y - [y] \). Thus, we can rewrite the function as:
\[ f(x) = \sin(\{x + 3\}) \]

### Step 2: Understand the properties of the fractional part function
The fractional part function \( \{x + 3\} \) is periodic with a period of 1. This means:
\[ \{x + 3\} = \{(x + 1) + 3\} = \{x + 3 + 1\} = \{x + 4\} \]
Thus, we can conclude that:
\[ \{x + 3\} = \{x\} \quad \text{(since adding an integer does not change the fractional part)} \]

### Step 3: Substitute back into the function
Now, we can rewrite the function as:
\[ f(x) = \sin(\{x\}) \]

### Step 4: Determine the period of the sine function
The function \( \sin(\{x\}) \) inherits the periodicity of the fractional part function. Since \( \{x\} \) is periodic with period 1, \( f(x) \) will also be periodic with period 1.

### Conclusion
Therefore, the period of the function \( f(x) = \sin(x + 3 - [x + 3]) \) is:
\[ \text{Period} = 1 \]

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