At which [tex]$x$[/tex]-values are the output values of the floor function [tex]$g(x)=\lfloor x\rfloor$[/tex] and the ceiling function [tex]$h(x)=\lceil x\rceil$[/tex] equal? Check all that apply.
-8
-5.2
-1.7
0
2.4
Answer (1)
The floor function ⌊ x ⌋ gives the greatest integer less than or equal to x , and the ceiling function ⌈ x ⌉ gives the smallest integer greater than or equal to x .
The floor and ceiling functions are equal if and only if x is an integer.
Check each given x -value to see if it is an integer.
The floor and ceiling functions are equal for x = − 8 and x = 0 , so the answer is − 8 , 0 .
Explanation
Understanding the Problem We are asked to find the x -values for which the floor function g ( x ) = ⌊ x ⌋ and the ceiling function h ( x ) = ⌈ x ⌉ are equal. The floor function gives the greatest integer less than or equal to x , while the ceiling function gives the smallest integer greater than or equal to x .
Key Concept The floor and ceiling functions are equal if and only if x is an integer. If x is an integer, then ⌊ x ⌋ = x and ⌈ x ⌉ = x , so ⌊ x ⌋ = ⌈ x ⌉ . If x is not an integer, then ⌊ x ⌋ < x < ⌈ x ⌉ , so ⌊ x ⌋ = ⌈ x ⌉ .
Checking Each Value We are given the following x values to check: -8, -5.2, -1.7, 0, 2.4.
Let's check each value:
For x = − 8 , since -8 is an integer, ⌊ − 8 ⌋ = − 8 and ⌈ − 8 ⌉ = − 8 , so ⌊ − 8 ⌋ = ⌈ − 8 ⌉ .
For x = − 5.2 , since -5.2 is not an integer, ⌊ − 5.2 ⌋ = − 6 and ⌈ − 5.2 ⌉ = − 5 , so ⌊ − 5.2 ⌋ = ⌈ − 5.2 ⌉ .
For x = − 1.7 , since -1.7 is not an integer, ⌊ − 1.7 ⌋ = − 2 and ⌈ − 1.7 ⌉ = − 1 , so ⌊ − 1.7 ⌋ = ⌈ − 1.7 ⌉ .
For x = 0 , since 0 is an integer, ⌊ 0 ⌋ = 0 and ⌈ 0 ⌉ = 0 , so ⌊ 0 ⌋ = ⌈ 0 ⌉ .
For x = 2.4 , since 2.4 is not an integer, ⌊ 2.4 ⌋ = 2 and ⌈ 2.4 ⌉ = 3 , so ⌊ 2.4 ⌋ = ⌈ 2.4 ⌉ .
Final Answer Therefore, the floor and ceiling functions are equal when x = − 8 and x = 0 .
Examples
The floor and ceiling functions are used in computer science to round numbers down or up to the nearest integer. For example, if you have 5.7 items and can only sell whole items, the floor function tells you that you can sell 5 items. The ceiling function tells you that you need to have 6 items in stock to fulfill a request for 5.7 items (since you can't sell a fraction of an item). These functions are also used in resource allocation, scheduling, and various other applications where discrete values are required.
