If [tex]f(x)=\frac{1}{2}\lfloor x\rfloor[/tex], what is [tex]f(7.6)[/tex]?
A. 3
B. 3.5
C. 3.8
D. 4
Answer (2)
Determine the floor of 7.6: ⌊ 7.6 ⌋ = 7 .
Substitute the floor value into the function: f ( 7.6 ) = 2 1 × 7 .
Calculate the result: f ( 7.6 ) = 3.5 .
The final answer is 3.5 .
Explanation
Understanding the problem We are given the function f ( x ) = 2 1 ⌊ x ⌋ and we need to find the value of f ( 7.6 ) . The floor function, denoted by ⌊ x ⌋ , returns the greatest integer less than or equal to x . In this case, we need to find the greatest integer less than or equal to 7.6.
Calculating the floor of 7.6 The greatest integer less than or equal to 7.6 is 7. Therefore, ⌊ 7.6 ⌋ = 7 .
Substituting into the function Now we substitute this value into the function: f ( 7.6 ) = 2 1 ⌊ 7.6 ⌋ = 2 1 × 7 = 3.5 .
Final Answer Therefore, f ( 7.6 ) = 3.5 .
Examples
The floor function is used in many real-life scenarios, such as determining the number of full boxes that can be filled with a certain number of items. For example, if you have 7.6 pounds of candy and each box holds 1 pound, you can fill 7 full boxes. Similarly, the function f ( x ) = 2 1 ⌊ x ⌋ could represent a scenario where you get half the value of the whole units. If x is the number of hours you worked and you get paid half the value of each full hour, then f ( x ) represents your earnings.
The value of the function f ( 7.6 ) is 3.5 after determining that ⌊ 7.6 ⌋ = 7 . Thus, f ( 7.6 ) = 2 1 × 7 . The correct answer is B. 3.5.
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