Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
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]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.