To determine the domain of the function [tex]\( y = 4 \lfloor x + 2 \rfloor \)[/tex], we need to consider the behavior of the floor function and the overall structure of the expression.
1. Understanding the Floor Function: The floor function, [tex]\(\lfloor x \rfloor\)[/tex], takes a real number [tex]\(x\)[/tex] and returns the largest integer less than or equal to [tex]\(x\)[/tex]. For any real number [tex]\(x\)[/tex], there is a corresponding value of [tex]\(\lfloor x \rfloor\)[/tex].
2. Inside the Floor Function: In the expression [tex]\( \lfloor x + 2 \rfloor \)[/tex], [tex]\(x\)[/tex] can be any real number. For any [tex]\(x\)[/tex], [tex]\(x + 2\)[/tex] will also be a real number, and thus [tex]\(\lfloor x + 2 \rfloor\)[/tex] will always be an integer.
3. Multiplying by 4: The expression [tex]\( 4 \lfloor x + 2 \rfloor \)[/tex] involves multiplying the integer from the floor function by 4. This means that for any given [tex]\(x\)[/tex], the function [tex]\( y = 4 \lfloor x + 2 \rfloor \)[/tex] will produce:
- An integer since the floor function output is an integer and multiplying an integer by 4 remains an integer.
4. Domain Consideration: The key point here is identifying the set of all possible values of [tex]\(x\)[/tex] for which [tex]\( y = 4 \lfloor x + 2 \rfloor \)[/tex] is defined. The floor function and addition operation inside it are defined for all real numbers [tex]\(x\)[/tex].
Therefore, the domain of [tex]\(y = 4 \lfloor x + 2 \rfloor\)[/tex] is all real numbers.
[tex]\(\boxed{\text{all real numbers}}\)[/tex]
