Given the function $y=\lfloor x\rfloor-1$, which values of the function are equivalent?
A. $f(-1)$ and $f(-2)$
B. $f(1)$ and $f(0)$
C. $f(-3)$ and $f(-2.1)$
D. $f(2)$ and $f(1.9)
Answer (2)
Evaluate f ( − 1 ) and f ( − 2 ) : f ( − 1 ) = − 2 , f ( − 2 ) = − 3 .
Evaluate f ( 1 ) and f ( 0 ) : f ( 1 ) = 0 , f ( 0 ) = − 1 .
Evaluate f ( − 3 ) and f ( − 2.1 ) : f ( − 3 ) = − 4 , f ( − 2.1 ) = − 4 .
Evaluate f ( 2 ) and f ( 1.9 ) : f ( 2 ) = 1 , f ( 1.9 ) = 0 .
The equivalent values are f ( − 3 ) and f ( − 2.1 ) .
f ( − 3 ) and f ( − 2.1 )
Explanation
Understanding the Problem We are given the function y = f ( x ) = ⌊ x ⌋ − 1 , where ⌊ x ⌋ represents the greatest integer less than or equal to x . We need to determine which of the given pairs of function values are equivalent.
Evaluating Function Values Let's evaluate each pair of function values:
f ( − 1 ) = ⌊ − 1 ⌋ − 1 = − 1 − 1 = − 2 f ( − 2 ) = ⌊ − 2 ⌋ − 1 = − 2 − 1 = − 3 Since − 2 = − 3 , f ( − 1 ) and f ( − 2 ) are not equivalent.
f ( 1 ) = ⌊ 1 ⌋ − 1 = 1 − 1 = 0 f ( 0 ) = ⌊ 0 ⌋ − 1 = 0 − 1 = − 1 Since 0 = − 1 , f ( 1 ) and f ( 0 ) are not equivalent.
f ( − 3 ) = ⌊ − 3 ⌋ − 1 = − 3 − 1 = − 4 f ( − 2.1 ) = ⌊ − 2.1 ⌋ − 1 = − 3 − 1 = − 4 Since − 4 = − 4 , f ( − 3 ) and f ( − 2.1 ) are equivalent.
f ( 2 ) = ⌊ 2 ⌋ − 1 = 2 − 1 = 1 f ( 1.9 ) = ⌊ 1.9 ⌋ − 1 = 1 − 1 = 0 Since 1 = 0 , f ( 2 ) and f ( 1.9 ) are not equivalent.
Identifying Equivalent Values Based on the calculations above, the only pair of function values that are equivalent is f ( − 3 ) and f ( − 2.1 ) .
Final Answer Therefore, the values of the function f ( − 3 ) and f ( − 2.1 ) are equivalent.
Examples
The floor function is used in many real-world applications, such as determining the number of full boxes that can be filled with a certain number of items. For example, if you have 10 apples and each box can hold 3 apples, then the number of full boxes you can fill is ⌊ 3 10 ⌋ = 3 . Understanding how the floor function works can help you solve problems involving discrete quantities and integer values. Another example is calculating shipping costs, where costs are often determined by the weight of the package rounded up to the nearest pound or kilogram.
The only pair of equivalent values from the function y = f ( x ) = ⌊ x ⌋ − 1 is f ( − 3 ) and f ( − 2.1 ) which both equal -4. Other pairs evaluated show they do not yield the same result. Hence, option C is the answer.
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