What are the domain and range of the step function with the equation below?
[tex]$f(x)=-3\lceil x\rceil$
A. The domain is all real numbers, and the range is all real numbers.
B. The domain is all integers, and the range is all integers that are multiples of 3.
C. The domain is all real numbers, and the range is all integers that are multiples of 3.
D. The domain is all integers, and the range is all real numbers.
Answer (2)
The domain of the function f ( x ) = − 3 ⌈ x ⌉ is all real numbers.
The ceiling function ⌈ x ⌉ returns an integer for any real number x .
Multiplying the ceiling function by -3 results in integer multiples of -3.
The range of f ( x ) is all integers that are multiples of -3, and the domain is all real numbers. Therefore, the answer is: The domain is all real numbers, and the range is all integers that are multiples of 3.
Explanation
Understanding the Problem We are given the step function f ( x ) = − 3 C e i l x . We need to determine its domain and range.
Determining the Domain The domain of the ceiling function C e i l x is all real numbers, since we can evaluate the ceiling function for any real number x . Multiplying by − 3 does not change the domain. Therefore, the domain of f ( x ) = − 3 C e i l x is all real numbers.
Determining the Range The range of the ceiling function C e i l x is the set of all integers. When we multiply the ceiling function by − 3 , we obtain − 3 C e i l x , where C e i l x is an integer. Thus, the range of f ( x ) is the set of all integers multiplied by − 3 . This is the same as the set of all integer multiples of − 3 , or equivalently, the set of all integer multiples of 3 with a negative sign.
Conclusion Therefore, the domain of f ( x ) is all real numbers, and the range is all integers that are multiples of − 3 .
Examples
Step functions are used in various real-world scenarios, such as modeling the cost of shipping based on weight. For example, if a shipping company charges a flat rate for each pound or fraction thereof, the cost can be modeled using a step function. Understanding the domain and range helps in determining the possible inputs (weights) and the corresponding outputs (shipping costs).
The domain of the function f ( x ) = − 3 ⌈ x ⌉ is all real numbers, and the range consists of all integers that are multiples of 3. Thus, the correct choice is option C. Both the ceiling function and the multiplication by -3 verify this result effectively.
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