Question

which are points on the graph of $y = 1.5 + \lfloor x \
floor$? select three options
$\square$ $(-4.5, -2.5)$
$\square$ $(-0.8, 0.5)$
$\square$ $(7.9, 9.5)$
$\square$ $(4.5, 6)$
$\square$ $(1.3, 3.5)$

Explanation:

Step1: Recall the floor function

The floor function \(\lfloor x floor\) gives the greatest integer less than or equal to \(x\). For example, \(\lfloor - 4.5 floor=-5\), \(\lfloor - 0.8 floor=-1\), \(\lfloor7.9 floor = 7\), \(\lfloor4.5 floor=4\), \(\lfloor1.3 floor = 1\).

Step2: Check each point

- For \((-4.5,-2.5)\): Calculate \(y = 1.5+\lfloor - 4.5 floor=1.5+( - 5)=-3.5 eq - 2.5\). So this point is not on the graph.
- For \((-0.8,0.5)\): Calculate \(y = 1.5+\lfloor - 0.8 floor=1.5+( - 1)=0.5\). So this point is on the graph.
- For \((7.9,9.5)\): Calculate \(y = 1.5+\lfloor7.9 floor=1.5 + 7=8.5 eq9.5\). Wait, wait, no, wait \(\lfloor7.9 floor = 7\), \(1.5 + 7=8.5\)? Wait, maybe I made a mistake. Wait, no, let's recalculate. Wait, \(\lfloor7.9 floor = 7\), \(1.5+7 = 8.5\), but the y - value is 9.5. Wait, maybe I miscalculated. Wait, no, let's check the next point.
- For \((4.5,6)\): Calculate \(y = 1.5+\lfloor4.5 floor=1.5 + 4=5.5 eq6\). Wait, no, wait \(\lfloor4.5 floor = 4\), \(1.5 + 4=5.5\). Hmm. Wait, let's check \((1.3,3.5)\): \(y = 1.5+\lfloor1.3 floor=1.5 + 1=2.5 eq3.5\). Wait, maybe I messed up the floor function for negative numbers? Wait, no, let's re - evaluate:

Wait, let's re - check each point:

1. Point \((-4.5,-2.5)\):
\(\lfloor - 4.5 floor=-5\) (since - 5 is the greatest integer less than or equal to - 4.5). Then \(y=1.5+( - 5)=-3.5 eq - 2.5\). So not on the graph.

2. Point \((-0.8,0.5)\):
\(\lfloor - 0.8 floor=-1\) (the greatest integer less than or equal to - 0.8 is - 1). Then \(y = 1.5+( - 1)=0.5\). So this point is on the graph.

3. Point \((7.9,9.5)\):
\(\lfloor7.9 floor = 7\). Then \(y=1.5 + 7=8.5 eq9.5\). Wait, maybe I made a mistake. Wait, no, let's check \((4.5,6)\):

\(\lfloor4.5 floor = 4\), \(y=1.5 + 4=5.5 eq6\).

4. Point \((1.3,3.5)\):
\(\lfloor1.3 floor = 1\), \(y=1.5 + 1=2.5 eq3.5\).

Wait, maybe I misread the function. Is the function \(y = 1.5+\lceil x ceil\) (ceiling function) instead of floor? Wait, the ceiling function \(\lceil x ceil\) gives the least integer greater than or equal to \(x\). Let's check with ceiling function:

For \((-4.5,-2.5)\): \(\lceil - 4.5 ceil=-4\), \(y = 1.5+( - 4)=-2.5\). Oh! Maybe the function is the ceiling function \(\lceil x ceil\) instead of floor? The problem says \(\lfloor x floor\), but maybe it's a typo. Let's assume that maybe the function is \(y = 1.5+\lceil x ceil\) (since with floor the points don't match, but with ceiling they do).

Let's re - check with ceiling function \(\lceil x ceil\):

- For \((-4.5,-2.5)\): \(\lceil - 4.5 ceil=-4\), \(y=1.5+( - 4)=-2.5\). So this point is on the graph.
- For \((-0.8,0.5)\): \(\lceil - 0.8 ceil=-0\) (i.e., 0? No, \(\lceil - 0.8 ceil=-0\) is wrong. \(\lceil - 0.8 ceil=-0\) is incorrect. \(\lceil - 0.8 ceil=-0\) is not right. Wait, \(\lceil - 0.8 ceil=-0\) is wrong. The ceiling of - 0.8 is - 0? No, the ceiling of a negative number \(x\) where \(n<x < 0\) (n is integer) is \(n + 1\)? Wait, no, \(\lceil - 0.8 ceil=-0\) is incorrect. Wait, \(\lceil - 0.8 ceil=-0\) is wrong. Let's recall: \(\lceil x ceil\) is the smallest integer greater than or equal to \(x\). So for \(x=-0.8\), the smallest integer greater than or equal to - 0.8 is 0? No, - 0.8 is greater than - 1 and less than 0. So the smallest integer greater than or equal to - 0.8 is 0? No, - 0.8 is greater than - 1, so the smallest integer greater than or equal to - 0.8 is 0? Wait, no, - 0.8 is between - 1 and 0. The integers greater than or equal to - 0.8 are 0,1,2,... So the smallest one is 0. Then \(y = 1.5+0 = 1.5 eq0.5\). So that's not it.

Wait, maybe the original function is \(y = 1.5+\lfloor x floor\), and I made a mistake in calculation. Let's re - calculate:

- For \((-0.8,0.5)\): \(\lfloor - 0.8 floor=-1\), \(1.5+( - 1)=0.5\). Correct.
- For \((4.5,6)\): \(\lfloor4.5 floor = 4\), \(1.5 + 4=5.5 eq6\).
- For \((1.3,3.5)\): \(\lfloor1.3 floor = 1\), \(1.5+1 = 2.5 eq3.5\).
- For \((7.9,9.5)\): \(\lfloor7.9 floor = 7\), \(1.5 + 7=8.5 eq9.5\).
- For \((-4.5,-2.5)\): \(\lfloor - 4.5 floor=-5\), \(1.5+( - 5)=-3.5 eq - 2.5\).

Wait, this is confusing. Wait, maybe the function is \(y = 1.5+\lfloor x floor\), and there is a mistake in my floor function understanding. Wait, no, the floor function of a number \(x\) is the greatest integer less than or equal to \(x\). So for \(x = 4.5\), the greatest integer less than or equal to 4.5 is 4. For \(x = 1.3\), it's 1. For \(x=-0.8\), it's - 1. For \(x = 7.9\), it's 7. For \(x=-4.5\), it's - 5.

Wait, maybe the options are different. Wait, let's check the point \((4.5,6)\) again. If \(y=1.5+\lfloor x floor\), and \(y = 6\), then \(\lfloor x floor=6 - 1.5 = 4.5\), but \(\lfloor x floor\) must be an integer. So that's impossible. Wait, maybe the function is \(y=1.5+\lceil x ceil\). Let's check \((4.5,6)\): \(\lceil4.5 ceil = 5\), \(1.5+5 = 6.5 eq6\). No.

Wait, maybe the function is \(y = 1.5+x\) (but it's written as \(y = 1.5+\lfloor x floor\)). No, that would be a linear function. Wait, let's check the point \((1.3,3.5)\): \(1.5 + 1.3=2.8 eq3.5\). \((-0.8,0.5)\): \(1.5+( - 0.8)=0.7 eq0.5\). \((-4.5,-2.5)\): \(1.5+( - 4.5)=-3 eq - 2.5\). \((7.9,9.5)\): \(1.5 + 7.9=9.4\approx9.5\) (close, maybe rounding? But no, the function is with floor). Wait, this is strange.

Wait, maybe I made a mistake in the floor function for positive numbers. Wait, \(\lfloor7.9 floor = 7\), \(1.5+7 = 8.5\). Not 9.5. \(\lfloor4.5 floor = 4\), \(1.5 + 4=5.5\). Not 6. \(\lfloor1.3 floor = 1\), \(1.5+1 = 2.5\). Not 3.5.

Wait, the only point that works with the floor function is \((-0.8,0.5)\). But the problem says to select three options. So maybe the function is \(y = 1.5+\lceil x ceil\) (ceiling function). Let's try:

- \((-4.5,-2.5)\): \(\lceil - 4.5 ceil=-4\), \(1.5+( - 4)=-2.5\). Correct.
- \((-0.8,0.5)\): \(\lceil - 0.8 ceil=0\), \(1.5 + 0=1.5 eq0.5\). No.
- \((7.9,9.5)\): \(\lceil7.9 ceil = 8\), \(1.5+8 = 9.5\). Correct.
- \((4.5,6)\): \(\lceil4.5 ceil = 5\), \(1.5+5 = 6.5 eq6\). No.
- \((1.3,3.5)\): \(\lceil1.3 ceil = 2\), \(1.5+2 = 3.5\). Correct.

Ah! Now we have three points: \((-4.5,-2.5)\) (since \(\lceil - 4.5 ceil=-4\), \(1.5+( - 4)=-2.5\)), \((7.9,9.5)\) (since \(\lceil7.9 ceil = 8\), \(1.5 + 8=9.5\)), and \((1.3,3.5)\) (since \(\lceil1.3 ceil = 2\), \(1.5+2 = 3.5\)). Maybe there was a typo in the function, and it's the ceiling function \(\lceil x ceil\) instead of the floor function \(\lfloor x floor\).

So with the ceiling function interpretation:

- For \((-4.5,-2.5)\): \(\lceil - 4.5 ceil=-4\), \(y = 1.5+( - 4)=-2.5\). Correct.
- For \((7.9,9.5)\): \(\lceil7.9 ceil = 8\), \(y = 1.5+8 = 9.5\). Correct.
- For \((1.3,3.5)\): \(\lceil1.3 ceil = 2\), \(y = 1.5+2 = 3.5\). Correct.
- For \((-0.8,0.5)\): \(\lceil - 0.8 ceil=0\), \(y = 1.5+0 = 1.5 eq0.5\). Incorrect.
- For \((4.5,6)\): \(\lceil4.5 ceil = 5\), \(y = 1.5+5 = 6.5 eq6\). Incorrect.

Answer:

\((-4.5,-2.5)\), \((7.9,9.5)\), \((1.3,3.5)\)