What is the domain of the step function [tex]f(x)=\lceil 2 x\rceil-1[/tex]?
A. [tex] \{x \mid x \geq-1\} [/tex]
B. [tex] \{x \mid x \geq 1\} [/tex]
C. [tex] \{x \mid x \text{ is an integer }\} [/tex]
D. [tex] \{x \mid x \text{ is a real number }\} [/tex]
Answer (1)
The function is f ( x ) = ⌈ 2 x ⌉ − 1 .
The ceiling function ⌈ x ⌉ is defined for all real numbers.
Therefore, ⌈ 2 x ⌉ is defined for all real numbers.
Subtracting 1 does not change the domain, so the domain of f ( x ) is all real numbers: x ∣ x is a real number .
Explanation
Understanding the Problem We are asked to find the domain of the function f ( x ) = ce i l 2 x ce i l − 1 . The domain of a function is the set of all possible values of x for which the function is defined.
Understanding the Ceiling Function The ceiling function, denoted by ⌈ x ⌉ , is defined for all real numbers x . It returns the smallest integer greater than or equal to x . For example, ⌈ 2.3 ⌉ = 3 , ⌈ 5 ⌉ = 5 , and ⌈ − 1.7 ⌉ = − 1 .
Analyzing the Function Since the ceiling function is defined for all real numbers, the expression ⌈ 2 x ⌉ is defined for all real numbers x . Multiplying x by 2 does not restrict the domain, and the ceiling function can be applied to any real number.
Determining the Domain The function f ( x ) = ⌈ 2 x ⌉ − 1 involves subtracting 1 from the result of the ceiling function. Subtracting a constant from a function does not change its domain. Therefore, the domain of f ( x ) is the same as the domain of ⌈ 2 x ⌉ , which is all real numbers.
Final Answer Therefore, the domain of the step function f ( x ) = ⌈ 2 x ⌉ − 1 is all real numbers.
Examples
Step functions are used in various real-world applications, such as modeling the cost of shipping based on weight. For example, the cost might be a fixed amount for weights up to a certain limit, then jump to a higher fixed amount for weights above that limit. This creates a step-like pattern. Another example is in digital signal processing, where continuous signals are converted into discrete steps for digital representation. Understanding the domain of step functions helps in analyzing the range of inputs for which these models are valid and applicable.
