12 multiple-choice questions, progressively harder.
Evaluate ⌈3.2⌉\lceil 3.2 \rceil⌈3.2⌉.
Solution
Correct answer: C
The ceiling is the least integer at or above the input.
3.2 lies between 3 and 4, so ⌈3.2⌉=43.2 \text{ lies between } 3 \text{ and } 4, \text{ so } \lceil 3.2 \rceil = 43.2 lies between 3 and 4, so ⌈3.2⌉=4
Rounding up from 3.23.23.2 gives 444.
Evaluate ⌊5⌋\lfloor 5 \rfloor⌊5⌋.
Correct answer: B
When the input is already an integer, the floor returns it unchanged.
⌊5⌋=5\lfloor 5 \rfloor = 5⌊5⌋=5
The number 555 is the greatest integer at or below itself.
Evaluate ⌈7⌉\lceil 7 \rceil⌈7⌉.
Correct answer: D
An integer input is left unchanged by the ceiling.
⌈7⌉=7\lceil 7 \rceil = 7⌈7⌉=7
Since 777 is already whole, the least integer at or above it is 777 itself.
Evaluate ⌊2.9⌋\lfloor 2.9 \rfloor⌊2.9⌋.
Correct answer: A
The floor rounds down, even when the input is close to the next integer.
2.9 lies between 2 and 3, so ⌊2.9⌋=22.9 \text{ lies between } 2 \text{ and } 3, \text{ so } \lfloor 2.9 \rfloor = 22.9 lies between 2 and 3, so ⌊2.9⌋=2
Being near 333 does not matter; the greatest integer at or below 2.92.92.9 is 222.
Evaluate ⌈−2.3⌉\lceil -2.3 \rceil⌈−2.3⌉.
−2.3 lies between −3 and −2, so ⌈−2.3⌉=−2-2.3 \text{ lies between } -3 \text{ and } -2, \text{ so } \lceil -2.3 \rceil = -2−2.3 lies between −3 and −2, so ⌈−2.3⌉=−2
The nearest integer to the right of −2.3-2.3−2.3 is −2-2−2.
What is the greatest integer less than or equal to 4.64.64.6?
This is the definition of the floor.
⌊4.6⌋=4\lfloor 4.6 \rfloor = 4⌊4.6⌋=4
The integers at or below 4.64.64.6 are 4,3,2,…4, 3, 2, \dots4,3,2,…, and the greatest is 444.
What is the least integer greater than or equal to 4.64.64.6?
This is the definition of the ceiling.
⌈4.6⌉=5\lceil 4.6 \rceil = 5⌈4.6⌉=5
The integers at or above 4.64.64.6 are 5,6,7,…5, 6, 7, \dots5,6,7,…, and the least is 555.
Evaluate ⌊−4⌋\lfloor -4 \rfloor⌊−4⌋.
When the input is already an integer, the floor returns it unchanged, negatives included.
⌊−4⌋=−4\lfloor -4 \rfloor = -4⌊−4⌋=−4
The greatest integer at or below −4-4−4 is −4-4−4 itself.
Evaluate ⌈−6⌉\lceil -6 \rceil⌈−6⌉.
An integer input is unchanged by the ceiling.
⌈−6⌉=−6\lceil -6 \rceil = -6⌈−6⌉=−6
The least integer at or above −6-6−6 is −6-6−6 itself.
What is the domain of f(x)=⌊x⌋f(x) = \lfloor x \rfloorf(x)=⌊x⌋?
Every real number can be rounded down to some integer, so nothing is excluded.
domain: all real x\text{domain: all real } xdomain: all real x
The floor accepts any input; only its outputs are restricted.
What is the range (set of outputs) of f(x)=⌊x⌋f(x) = \lfloor x \rfloorf(x)=⌊x⌋?
The floor always returns a whole number, and every integer is reached.
range: the integers\text{range: the integers}range: the integers
For example ⌊3.7⌋=3\lfloor 3.7 \rfloor = 3⌊3.7⌋=3 and ⌊−2.3⌋=−3\lfloor -2.3 \rfloor = -3⌊−2.3⌋=−3 are both integers.
The graph of y=⌊x⌋y = \lfloor x \rfloory=⌊x⌋ has what shape?
On each interval from an integer nnn up to n+1n + 1n+1, the floor holds the constant value nnn, then jumps up by one.
⌊x⌋=n for n≤x<n+1\lfloor x \rfloor = n \text{ for } n \le x < n + 1⌊x⌋=n for n≤x<n+1
Those flat pieces joined by jumps make a staircase, a step function.
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