Which points lie on the graph of the function [tex]f(x)=\lceil x\rceil+2[/tex]? Check all that apply.
[tex](-5.5,-4)[/tex]
[tex](-3.8,-2)[/tex]
[tex](-1.1, 1)[/tex]
[tex](-0.9, 2)[/tex]
[tex](2.2,5)[/tex]
[tex](4.7,6)[/tex]
Answer (1)
Calculate the ceiling of the x-coordinate for each point.
Add 2 to the result of the ceiling function.
Check if the calculated value matches the y-coordinate of the point.
The points that satisfy the equation y = ⌈ x ⌉ + 2 are ( − 1.1 , 1 ) , ( − 0.9 , 2 ) , and ( 2.2 , 5 ) .
The points that lie on the graph are: ( − 1.1 , 1 ) , ( − 0.9 , 2 ) , ( 2.2 , 5 ) .
Explanation
Understanding the Problem We are given the function f ( x ) = ⌈ x ⌉ + 2 , where ⌈ x ⌉ represents the ceiling function (the smallest integer greater than or equal to x ). We need to determine which of the given points lie on the graph of this function. A point ( x , y ) lies on the graph if and only if y = f ( x ) = ⌈ x ⌉ + 2 .
Checking Each Point Let's check each point:
Point (-5.5, -4): ⌈ − 5.5 ⌉ = − 5 . So, f ( − 5.5 ) = − 5 + 2 = − 3 . Since − 3 = − 4 , this point does not lie on the graph.
Point (-3.8, -2): ⌈ − 3.8 ⌉ = − 3 . So, f ( − 3.8 ) = − 3 + 2 = − 1 . Since − 1 = − 2 , this point does not lie on the graph.
Point (-1.1, 1): ⌈ − 1.1 ⌉ = − 1 . So, f ( − 1.1 ) = − 1 + 2 = 1 . Since 1 = 1 , this point lies on the graph.
Point (-0.9, 2): ⌈ − 0.9 ⌉ = 0 . So, f ( − 0.9 ) = 0 + 2 = 2 . Since 2 = 2 , this point lies on the graph.
Point (2.2, 5): ⌈ 2.2 ⌉ = 3 . So, f ( 2.2 ) = 3 + 2 = 5 . Since 5 = 5 , this point lies on the graph.
Point (4.7, 6): ⌈ 4.7 ⌉ = 5 . So, f ( 4.7 ) = 5 + 2 = 7 . Since 7 = 6 , this point does not lie on the graph.
Final Answer Therefore, the points that lie on the graph of the function f ( x ) = ⌈ x ⌉ + 2 are ( − 1.1 , 1 ) , ( − 0.9 , 2 ) , and ( 2.2 , 5 ) .
Examples
Imagine you're designing a staircase where each step's height is determined by the ceiling function plus a constant. If x represents the desired step number, f ( x ) = ⌈ x ⌉ + 2 could model the actual height. Checking points on the graph helps ensure the staircase meets specific height requirements at certain steps. This concept applies to various scenarios, like setting prices based on tiered usage or calculating resource allocation in discrete units, where the ceiling function ensures you always round up to the nearest whole unit.
