The period of the function f(x)= sin(x+3-[x+3]), where [ ] denotes the greatest integer function is

To determine the period of the function \( f(x) = \sin(x + 3 - [x + 3]) \), where \([x + 3]\) represents the greatest integer function (also known as the floor function), we need to break down the components of the function step by step.

Understanding the Components

The expression \( x + 3 - [x + 3] \) can be simplified. The term \([x + 3]\) gives us the greatest integer less than or equal to \( x + 3 \). Therefore, \( x + 3 - [x + 3] \) represents the fractional part of \( x + 3 \). This can be expressed as:

Fractional Part: \( \{x + 3\} = x + 3 - [x + 3] \)

Behavior of the Fractional Part

The fractional part \( \{x + 3\} \) always lies in the interval \([0, 1)\). As \( x \) increases, \( \{x + 3\} \) resets to 0 every time \( x + 3 \) reaches a new integer. This means that the function \( \{x + 3\} \) is periodic with a period of 1, since it repeats its values every time \( x \) increases by 1.

Analyzing the Sine Function

Now, we can substitute this back into our original function:

Function Simplification: \( f(x) = \sin(\{x + 3\}) \)

Since the sine function itself has a period of \( 2\pi \), we need to consider how the periodicity of the fractional part affects the overall function. The sine function will take the same values for any input that differs by an integer multiple of \( 2\pi \). However, because \( \{x + 3\} \) only varies between 0 and 1, we are only interested in how \( \sin(\{x + 3\}) \) behaves within that interval.

Finding the Period of the Combined Function

Given that \( \{x + 3\} \) has a period of 1, we can conclude that:

Period of \( f(x) \): The function \( f(x) = \sin(\{x + 3\}) \) inherits this periodicity, resulting in a period of 1.

Conclusion

In summary, the period of the function \( f(x) = \sin(x + 3 - [x + 3]) \) is 1. This is due to the periodic nature of the fractional part, which resets every time \( x \) increases by 1, and the sine function's behavior within that interval. Thus, the final answer is:

Period of \( f(x) \): 1