Answer :
To determine the domain of the step function [tex]\( f(x) = \lceil 2x \rceil - 1 \)[/tex], we need to analyze the components of the function and understand where it is defined.
1. Understanding [tex]\( \lceil 2x \rceil \)[/tex]:
- The function [tex]\( 2x \)[/tex] is simply multiplying [tex]\( x \)[/tex] by 2, which is an operation defined for all real numbers.
- The ceiling function [tex]\( \lceil y \rceil \)[/tex] returns the smallest integer greater than or equal to [tex]\( y \)[/tex]. This function is defined for all real numbers [tex]\( y \)[/tex].
2. Combining the components:
- Since [tex]\( 2x \)[/tex] is defined for all [tex]\( x \in \mathbb{R} \)[/tex], the expression [tex]\( 2x \)[/tex] will always yield a real number.
- Applying the ceiling function to this real number, [tex]\( \lceil 2x \rceil \)[/tex], produces an integer.
- Subtracting 1 from this integer will result in another integer, ensuring that [tex]\( f(x) \)[/tex] will produce an integer for any real number [tex]\( x \)[/tex].
3. Conclusion:
- Since the function [tex]\( \lceil 2x \rceil \)[/tex] is valid for all [tex]\( x \in \mathbb{R} \)[/tex], the function [tex]\( f(x) \)[/tex] will also be defined for all real numbers [tex]\( x \)[/tex].
Thus, the domain of the function [tex]\( f(x) = \lceil 2x \rceil - 1 \)[/tex] is all real numbers, and our answer is:
[tex]\[ \boxed{\{x \mid x \text{ is a real number} \}} \][/tex]
1. Understanding [tex]\( \lceil 2x \rceil \)[/tex]:
- The function [tex]\( 2x \)[/tex] is simply multiplying [tex]\( x \)[/tex] by 2, which is an operation defined for all real numbers.
- The ceiling function [tex]\( \lceil y \rceil \)[/tex] returns the smallest integer greater than or equal to [tex]\( y \)[/tex]. This function is defined for all real numbers [tex]\( y \)[/tex].
2. Combining the components:
- Since [tex]\( 2x \)[/tex] is defined for all [tex]\( x \in \mathbb{R} \)[/tex], the expression [tex]\( 2x \)[/tex] will always yield a real number.
- Applying the ceiling function to this real number, [tex]\( \lceil 2x \rceil \)[/tex], produces an integer.
- Subtracting 1 from this integer will result in another integer, ensuring that [tex]\( f(x) \)[/tex] will produce an integer for any real number [tex]\( x \)[/tex].
3. Conclusion:
- Since the function [tex]\( \lceil 2x \rceil \)[/tex] is valid for all [tex]\( x \in \mathbb{R} \)[/tex], the function [tex]\( f(x) \)[/tex] will also be defined for all real numbers [tex]\( x \)[/tex].
Thus, the domain of the function [tex]\( f(x) = \lceil 2x \rceil - 1 \)[/tex] is all real numbers, and our answer is:
[tex]\[ \boxed{\{x \mid x \text{ is a real number} \}} \][/tex]
