Points lie on the graph of the function [tex]f(x)=\lceil x\rceil+2[/tex]? Check all that apply.
(-5.5, -4)
(-3.8, -2)
(-1.1, 1)
(-0.9, 2)
(2.2, 5)
(4.7, 6)
Answer (2)
The function is f ( x ) = ⌈ x ⌉ + 2 .
Evaluate f ( x ) for each given x -coordinate.
Check if f ( x ) equals the corresponding y -coordinate.
The points that lie on the graph are ( − 1.1 , 1 ) , ( − 0.9 , 2 ) , and ( 2.2 , 5 ) . ( − 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 f ( x ) = y .
Evaluating Each Point Let's evaluate each point:
(-5.5, -4): f ( − 5.5 ) = ⌈ − 5.5 ⌉ + 2 = − 5 + 2 = − 3 . Since − 3 = − 4 , this point does not lie on the graph.
(-3.8, -2): f ( − 3.8 ) = ⌈ − 3.8 ⌉ + 2 = − 3 + 2 = − 1 . Since − 1 = − 2 , this point does not lie on the graph.
(-1.1, 1): f ( − 1.1 ) = ⌈ − 1.1 ⌉ + 2 = − 1 + 2 = 1 . Since 1 = 1 , this point lies on the graph.
(-0.9, 2): f ( − 0.9 ) = ⌈ − 0.9 ⌉ + 2 = 0 + 2 = 2 . Since 2 = 2 , this point lies on the graph.
(2.2, 5): f ( 2.2 ) = ⌈ 2.2 ⌉ + 2 = 3 + 2 = 5 . Since 5 = 5 , this point lies on the graph.
(4.7, 6): f ( 4.7 ) = ⌈ 4.7 ⌉ + 2 = 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
The ceiling function is used in many real-world applications, such as determining the number of boxes needed to ship a certain number of items. For example, if you have 10.5 items to ship and each box can hold only one item, you would need ⌈ 10.5 ⌉ = 11 boxes. Similarly, if you are calculating the cost of parking in a garage that charges by the hour, the ceiling function can be used to determine the number of hours you will be charged for, even if you only park for a fraction of an hour. Understanding functions like ceiling function helps us to model and solve problems involving discrete quantities and rates.
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).
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