I need help on this math question please.

Question

Examine the following step function.
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Which statements are true? Select all that apply

Any decimal or fraction rounds to the greatest integer that is smaller than x

It is the ceiling function.

It is the floor function.

Any decimal or fraction rounds to the least integer that is greater than x.

Answer 1

Hi Nathan,

An open circle means that all the numbers on that line segment exist except for that point. If you have an open circle at (-1,0), that means that your line segment contains numbers like

(-0.9,0)

(-0.9999,0)

(-0.99999999999999,0)

But not (-1,0).

A closed circle means your line segment DOES in fact contain that point. Since your circle is closed at (0,0), that means your function exists at (0,0).

That line segment, therefore, includes all numbers from

(-0.99999999999999, 0) to (0,0).

The 9's in that first point go on forever.

Just above the closed circle at the origin, you have an open circle at (0,1). This connects a line segment to a closed circle at (1,1). Similar to last time, that means the line segment contains values very close to zero but just above zero, like

(0.1, 1)

(0.00001, 1)

(0.0000000000000000000001, 1)

This upper line segment, however, does NOT contain (0,1). Once your x-value is actually equal to 0, the closed circle on the segment below it takes over and the point is (0,0) instead of (0,1).

That's the behavior of your function. What's a good way to think of what's going on?

If you plug in an x-value like 0.8, what will your y-value be? An x-value of 0.8 would land on the line segment in between the open circle at (0,1) and the closed circle at (1,1). In fact, any value between 0 (non-inclusive) and 1 (inclusive) would be 1.

This is true for all values along this segment. This function turns

0.1 --> 1

0.4 --> 1

0.9 --> 1

0.00000001 --> 1

1 --> 1

So anything that's even barely above 0 gets bumped up to 1.

The pattern repeats for the other segments along your graph as well. As long as you're even a tiny bit above a whole number, you get bumped up to the next whole number.

1.00001 --> 2

1.9999 --> 2

2 --> 2

2.1 -- > 3

5.0001 --> 6

5.9 --> 6

5.9999999 --> 6

6 --> 6

6.1 --> 7

This is what we call a "ceiling function". It ignores typical rounding rules and decides everything just gets rounded UP, so even 2.1 becomes 3. It's how phone companies used to charge for minutes before unlimited talk became a thing. If your call was 22 minutes and 3 seconds they charged for 23 minutes.

This is different from what we call a "floor function". This one ignores rounding rules and decides every decimal gets rounded DOWN to the nearest whole number, so 2.9 becomes 2. This is how age is decided. If your 21th birthday is tomorrow, you're technically 20 years and 364 days old (20.99726 years old). Despite that, you're still considered 20 and a bar will not sell you a beer.

You have two correct statements:

1) It is a ceiling function. This is because every decimal gets rounded up to its nearest whole number.

2) Any decimal or fraction rounds to the least integer that is greater than x. This is the definition of ceiling function.

Hope this helps.

Answer 2

This might help:

A ceiling function returns the smallest nearest integer which is greater than or equal to the specified number.

The floor function returns the largest nearest integer which is less than or equal to a specified value.

For example, take the number 3.3. The ceiling of this number is 4, the floor of this number is 3.