The floor function returns the largest integer that does not exceed the input. The integers less than −2.7 are …, −5, −4, −3, and the largest among them is −3. Many students incorrectly choose −2, applying the intuition of 'rounding toward zero.' But the floor function always rounds toward negative infinity, so for negative numbers it goes further negative, not closer to zero.
Question 2 Multiple Choice
A parking garage charges $6 for each full or partial hour. A driver parks for 1 hour and 25 minutes. What is the correct charge?
A$6.00 — they've been there less than 1.5 hours, so the first-hour rate applies
B$8.50 — multiply the hourly rate by 1.42 hours
C$12.00 — 1 hour and any partial additional hour means they are charged for 2 hours
D$9.00 — round to the nearest hour
This is a ceiling-function scenario: ⌈1.42⌉ = 2, so the driver is charged for 2 full hours at $6 = $12. Step functions model discrete jumps, not continuous scaling. The charge stays at $6 for all of [0,1) hours, then jumps to $12 for all of [1,2) hours. A driver parked 1 hour 1 minute pays exactly the same as one parked 1 hour 59 minutes.
Question 3 True / False
The floor function ⌊x⌋ rounds any non-integer input toward the nearest integer.
TTrue
FFalse
Answer: False
The floor function always rounds toward negative infinity, not toward the nearest integer. For positive numbers, 'toward negative infinity' and 'toward zero' look the same (⌊3.7⌋ = 3, and 3 is both the nearest integer below and nearer to 0). But for negative numbers they diverge: ⌊−1.2⌋ = −2 (toward negative infinity), not −1 (the nearest integer). The ceiling function ⌈x⌉ rounds toward positive infinity.
Question 4 True / False
On the graph of f(x) = ⌊x⌋, the horizontal segment covering the interval [3, 4) has a closed circle at x = 3 and an open circle at x = 4.
TTrue
FFalse
Answer: True
Each piece of the floor function is defined on an interval of the form [n, n+1) — closed on the left, open on the right. At x = 3, the output is 3 and x = 3 IS included (closed circle). At x = 4, the output would jump to 4, so x = 4 is NOT part of this piece (open circle). This open-left/closed-right pattern is directly inherited from the piecewise definition.
Question 5 Short Answer
Why does ⌊−1.2⌋ equal −2 rather than −1, and how would you explain the difference to someone applying 'round toward zero'?
Think about your answer, then reveal below.
Model answer: The floor function returns the largest integer that does not exceed x. −1 does not satisfy this criterion for −1.2 because −1 > −1.2, meaning −1 exceeds the input. The largest integer that is ≤ −1.2 is −2. The 'round toward zero' rule is a mistake imported from ordinary rounding: it produces −1 because −1 is between 0 and −1.2, but the floor function doesn't care about closeness to zero — it cares about not exceeding the input.
The key conceptual test is: does the candidate integer exceed x? −1 exceeds −1.2 (since −1 > −1.2 on the number line), so it fails. −2 does not exceed −1.2 (since −2 < −1.2), so it qualifies. The floor function is essentially asking: 'what integer am I standing on if I walk down the number line from x and stop at the first integer I reach?'