Answer :
Sure, let's determine which points lie on the graph of the function [tex]\( f(x) = \lfloor x \rfloor + 2 \)[/tex], where [tex]\( \lfloor x \rfloor \)[/tex] represents the floor function, which gives the greatest integer less than or equal to [tex]\( x \)[/tex].
Step 1: Evaluate the function at given points.
For each point [tex]\((x, y)\)[/tex], we will calculate [tex]\( f(x) \)[/tex] and see if it matches [tex]\( y \)[/tex].
1. Point [tex]\((-5.5, -4)\)[/tex]:
[tex]\[ f(-5.5) = \lfloor -5.5 \rfloor + 2 \][/tex]
[tex]\[ \lfloor -5.5 \rfloor = -6 \quad \text{(since -6 is the greatest integer less than -5.5)} \][/tex]
[tex]\[ f(-5.5) = -6 + 2 = -4 \][/tex]
[tex]\[ \text{This point \((-5.5, -4)\) lies on the graph.} \][/tex]
2. Point [tex]\((-3.8, -2)\)[/tex]:
[tex]\[ f(-3.8) = \lfloor -3.8 \rfloor + 2 \][/tex]
[tex]\[ \lfloor -3.8 \rfloor = -4 \quad \text{(since -4 is the greatest integer less than -3.8)} \][/tex]
[tex]\[ f(-3.8) = -4 + 2 = -2 \][/tex]
[tex]\[ \text{This point \((-3.8, -2)\) lies on the graph.} \][/tex]
3. Point [tex]\((-1.1, 1)\)[/tex]:
[tex]\[ f(-1.1) = \lfloor -1.1 \rfloor + 2 \][/tex]
[tex]\[ \lfloor -1.1 \rfloor = -2 \quad \text{(since -2 is the greatest integer less than -1.1)} \][/tex]
[tex]\[ f(-1.1) = -2 + 2 = 0 \][/tex]
[tex]\[ \text{This point \((-1.1, 1)\) does not lie on the graph.} \][/tex]
4. Point [tex]\((-0.9, 2)\)[/tex]:
[tex]\[ f(-0.9) = \lfloor -0.9 \rfloor + 2 \][/tex]
[tex]\[ \lfloor -0.9 \rfloor = -1 \quad \text{(since -1 is the greatest integer less than -0.9)} \][/tex]
[tex]\[ f(-0.9) = -1 + 2 = 1 \][/tex]
[tex]\[ \text{This point \((-0.9, 2)\) does not lie on the graph.} \][/tex]
5. Point [tex]\((2.2, 5)\)[/tex]:
[tex]\[ f(2.2) = \lfloor 2.2 \rfloor + 2 \][/tex]
[tex]\[ \lfloor 2.2 \rfloor = 2 \quad \text{(since 2 is the greatest integer less than 2.2)} \][/tex]
[tex]\[ f(2.2) = 2 + 2 = 4 \][/tex]
[tex]\[ \text{This point \((2.2, 5)\) does not lie on the graph.} \][/tex]
6. Point [tex]\((4.7, 6)\)[/tex]:
[tex]\[ f(4.7) = \lfloor 4.7 \rfloor + 2 \][/tex]
[tex]\[ \lfloor 4.7 \rfloor = 4 \quad \text{(since 4 is the greatest integer less than 4.7)} \][/tex]
[tex]\[ f(4.7) = 4 + 2 = 6 \][/tex]
[tex]\[ \text{This point \((4.7, 6)\) lies on the graph.} \][/tex]
Step 2: List the points that lie on the graph.
After evaluating all the points, we find that the following points lie on the graph of the function [tex]\( f(x) = \lfloor x \rfloor + 2 \)[/tex]:
[tex]\[ (-5.5, -4), (-3.8, -2), (4.7, 6) \][/tex]
Step 1: Evaluate the function at given points.
For each point [tex]\((x, y)\)[/tex], we will calculate [tex]\( f(x) \)[/tex] and see if it matches [tex]\( y \)[/tex].
1. Point [tex]\((-5.5, -4)\)[/tex]:
[tex]\[ f(-5.5) = \lfloor -5.5 \rfloor + 2 \][/tex]
[tex]\[ \lfloor -5.5 \rfloor = -6 \quad \text{(since -6 is the greatest integer less than -5.5)} \][/tex]
[tex]\[ f(-5.5) = -6 + 2 = -4 \][/tex]
[tex]\[ \text{This point \((-5.5, -4)\) lies on the graph.} \][/tex]
2. Point [tex]\((-3.8, -2)\)[/tex]:
[tex]\[ f(-3.8) = \lfloor -3.8 \rfloor + 2 \][/tex]
[tex]\[ \lfloor -3.8 \rfloor = -4 \quad \text{(since -4 is the greatest integer less than -3.8)} \][/tex]
[tex]\[ f(-3.8) = -4 + 2 = -2 \][/tex]
[tex]\[ \text{This point \((-3.8, -2)\) lies on the graph.} \][/tex]
3. Point [tex]\((-1.1, 1)\)[/tex]:
[tex]\[ f(-1.1) = \lfloor -1.1 \rfloor + 2 \][/tex]
[tex]\[ \lfloor -1.1 \rfloor = -2 \quad \text{(since -2 is the greatest integer less than -1.1)} \][/tex]
[tex]\[ f(-1.1) = -2 + 2 = 0 \][/tex]
[tex]\[ \text{This point \((-1.1, 1)\) does not lie on the graph.} \][/tex]
4. Point [tex]\((-0.9, 2)\)[/tex]:
[tex]\[ f(-0.9) = \lfloor -0.9 \rfloor + 2 \][/tex]
[tex]\[ \lfloor -0.9 \rfloor = -1 \quad \text{(since -1 is the greatest integer less than -0.9)} \][/tex]
[tex]\[ f(-0.9) = -1 + 2 = 1 \][/tex]
[tex]\[ \text{This point \((-0.9, 2)\) does not lie on the graph.} \][/tex]
5. Point [tex]\((2.2, 5)\)[/tex]:
[tex]\[ f(2.2) = \lfloor 2.2 \rfloor + 2 \][/tex]
[tex]\[ \lfloor 2.2 \rfloor = 2 \quad \text{(since 2 is the greatest integer less than 2.2)} \][/tex]
[tex]\[ f(2.2) = 2 + 2 = 4 \][/tex]
[tex]\[ \text{This point \((2.2, 5)\) does not lie on the graph.} \][/tex]
6. Point [tex]\((4.7, 6)\)[/tex]:
[tex]\[ f(4.7) = \lfloor 4.7 \rfloor + 2 \][/tex]
[tex]\[ \lfloor 4.7 \rfloor = 4 \quad \text{(since 4 is the greatest integer less than 4.7)} \][/tex]
[tex]\[ f(4.7) = 4 + 2 = 6 \][/tex]
[tex]\[ \text{This point \((4.7, 6)\) lies on the graph.} \][/tex]
Step 2: List the points that lie on the graph.
After evaluating all the points, we find that the following points lie on the graph of the function [tex]\( f(x) = \lfloor x \rfloor + 2 \)[/tex]:
[tex]\[ (-5.5, -4), (-3.8, -2), (4.7, 6) \][/tex]
