12 multiple-choice questions, progressively harder.
Evaluate ⌊−5.5⌋\lfloor -5.5 \rfloor⌊−5.5⌋.
Solution
Correct answer: B
The floor rounds down to the nearest integer to the left.
−5.5 lies between −6 and −5, so ⌊−5.5⌋=−6-5.5 \text{ lies between } -6 \text{ and } -5, \text{ so } \lfloor -5.5 \rfloor = -6−5.5 lies between −6 and −5, so ⌊−5.5⌋=−6
Rounding down from −5.5-5.5−5.5 moves away from zero to −6-6−6.
Evaluate ⌈−5.5⌉\lceil -5.5 \rceil⌈−5.5⌉.
Correct answer: D
The ceiling is the nearest integer to the right.
−5.5 lies between −6 and −5, so ⌈−5.5⌉=−5-5.5 \text{ lies between } -6 \text{ and } -5, \text{ so } \lceil -5.5 \rceil = -5−5.5 lies between −6 and −5, so ⌈−5.5⌉=−5
Rounding up from −5.5-5.5−5.5 moves toward zero to −5-5−5.
Compute the fractional part {−3.4}\{-3.4\}{−3.4}, where {x}=x−⌊x⌋\{x\} = x - \lfloor x \rfloor{x}=x−⌊x⌋.
The floor of −3.4-3.4−3.4 rounds down to −4-4−4.
{−3.4}=−3.4−⌊−3.4⌋=−3.4−(−4)=0.6\{-3.4\} = -3.4 - \lfloor -3.4 \rfloor = -3.4 - (-4) = 0.6{−3.4}=−3.4−⌊−3.4⌋=−3.4−(−4)=0.6
The fractional part is never negative, so it lands in [0,1)[0, 1)[0,1).
Evaluate ⌊−10⌋\lfloor -10 \rfloor⌊−10⌋.
Correct answer: C
An integer input is returned unchanged by the floor.
⌊−10⌋=−10\lfloor -10 \rfloor = -10⌊−10⌋=−10
The greatest integer at or below −10-10−10 is −10-10−10 itself.
Using the identity ⌈x⌉=−⌊−x⌋\lceil x \rceil = -\lfloor -x \rfloor⌈x⌉=−⌊−x⌋, evaluate ⌈−2.7⌉\lceil -2.7 \rceil⌈−2.7⌉.
Correct answer: A
Negate the input, take the floor, then flip the sign.
⌈−2.7⌉=−⌊2.7⌋=−2\lceil -2.7 \rceil = -\lfloor 2.7 \rfloor = -2⌈−2.7⌉=−⌊2.7⌋=−2
This matches rounding −2.7-2.7−2.7 up to −2-2−2 directly.
A farm packs 535353 eggs into cartons of 121212. How many cartons are needed to hold them all?
Every leftover egg still needs a carton, so round up with a ceiling.
⌈53/12⌉=⌈4.41…⌉=5\lceil 53 / 12 \rceil = \lceil 4.41\ldots \rceil = 5⌈53/12⌉=⌈4.41…⌉=5
Four cartons hold 484848 eggs; the last 555 need a fifth carton.
Using the same rule ⌊x+12⌋\lfloor x + \tfrac{1}{2} \rfloor⌊x+21⌋, round −3.5-3.5−3.5 to the nearest integer.
Add one half, then take the floor.
⌊−3.5+12⌋=⌊−3⌋=−3\lfloor -3.5 + \tfrac{1}{2} \rfloor = \lfloor -3 \rfloor = -3⌊−3.5+21⌋=⌊−3⌋=−3
The half-up rule always breaks the tie upward (toward positive infinity), sending −3.5-3.5−3.5 to −3-3−3, just as it sent 3.53.53.5 up to 444.
If xxx is not an integer, what is ⌈x⌉−⌊x⌋\lceil x \rceil - \lfloor x \rfloor⌈x⌉−⌊x⌋?
A non-integer sits strictly between two consecutive integers, so its ceiling is one above its floor.
⌈x⌉−⌊x⌋=1\lceil x \rceil - \lfloor x \rfloor = 1⌈x⌉−⌊x⌋=1
The difference is 000 only on the integers, where the two functions agree.
Evaluate ⌊3.7⌋⋅⌈−1.2⌉\lfloor 3.7 \rfloor \cdot \lceil -1.2 \rceil⌊3.7⌋⋅⌈−1.2⌉.
Evaluate each bracket first, then multiply.
⌊3.7⌋⋅⌈−1.2⌉=3⋅(−1)=−3\lfloor 3.7 \rfloor \cdot \lceil -1.2 \rceil = 3 \cdot (-1) = -3⌊3.7⌋⋅⌈−1.2⌉=3⋅(−1)=−3
The ceiling of −1.2-1.2−1.2 is −1-1−1, the nearest integer to its right.
For which of these values does ⌊x⌋=5\lfloor x \rfloor = 5⌊x⌋=5?
The floor equals 555 exactly on 5≤x<65 \le x < 65≤x<6.
5≤5.5<6, so ⌊5.5⌋=55 \le 5.5 < 6, \text{ so } \lfloor 5.5 \rfloor = 55≤5.5<6, so ⌊5.5⌋=5
The others give ⌊4.9⌋=4\lfloor 4.9 \rfloor = 4⌊4.9⌋=4, ⌊6.0⌋=6\lfloor 6.0 \rfloor = 6⌊6.0⌋=6, and ⌊4.5⌋=4\lfloor 4.5 \rfloor = 4⌊4.5⌋=4.
Evaluate ⌊−0.5⌋\lfloor -0.5 \rfloor⌊−0.5⌋.
−0.5 lies between −1 and 0, so ⌊−0.5⌋=−1-0.5 \text{ lies between } -1 \text{ and } 0, \text{ so } \lfloor -0.5 \rfloor = -1−0.5 lies between −1 and 0, so ⌊−0.5⌋=−1
Rounding down from −0.5-0.5−0.5 moves away from zero to −1-1−1.
What is the range (set of outputs) of g(x)=⌈x⌉g(x) = \lceil x \rceilg(x)=⌈x⌉?
The ceiling always returns a whole number, and every integer is reached.
range: the integers\text{range: the integers}range: the integers
The domain is all real numbers, but the outputs are only ever integers.
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