To determine the domain of the step function [tex]\( f(x) = \lceil 2x \rceil - 1 \)[/tex], let's analyze it step by step.
1. Understanding the Ceiling Function:
The ceiling function [tex]\(\lceil y \rceil\)[/tex] takes any real number [tex]\(y\)[/tex] and rounds it up to the nearest integer. For example:
- [tex]\(\lceil 3.2 \rceil = 4\)[/tex]
- [tex]\(\lceil -1.5 \rceil = -1\)[/tex]
2. Applying the Ceiling Function:
In the function [tex]\( f(x) = \lceil 2x \rceil - 1 \)[/tex], we first compute [tex]\( 2x \)[/tex] and then apply the ceiling function to it, and finally subtract 1.
- If [tex]\( x = 0.5 \)[/tex], then [tex]\( 2x = 1 \)[/tex], and [tex]\(\lceil 1 \rceil = 1\)[/tex], so [tex]\( f(0.5) = 1 - 1 = 0\)[/tex].
- If [tex]\( x = -1.5 \)[/tex], then [tex]\( 2x = -3 \)[/tex], and [tex]\(\lceil -3 \rceil = -3\)[/tex], so [tex]\( f(-1.5) = -3 - 1 = -4 \)[/tex].
3. Identifying the Domain:
The domain of a function is the set of all input values [tex]\( x \)[/tex] for which the function is defined.
- The expression [tex]\( 2x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
- The ceiling function [tex]\(\lceil 2x \rceil \)[/tex] is defined for all real numbers [tex]\( 2x \)[/tex].
- Subtracting 1 to [tex]\(\lceil 2x \rceil - 1\)[/tex] is also defined for all real numbers.
Since there are no restrictions on [tex]\( x \)[/tex] in the function [tex]\( f(x) = \lceil 2x \rceil - 1 \)[/tex], the domain of [tex]\( f(x) \)[/tex] is all real numbers.
Hence, the domain of the function [tex]\( f(x) \)[/tex] is:
[tex]\[
\{ x \mid x \text{ is a real number} \}
\][/tex]
So, the correct option is:
[tex]\[
\{ x \mid x \text{ is a real number} \}
\][/tex]
