Answer :
To determine which points lie on the graph of the function [tex]\( f(x) = \lfloor x \rfloor + 2 \)[/tex], we must check each provided point to see if it satisfies the function. Note that [tex]\( \lfloor x \rfloor \)[/tex] denotes the floor function, which returns the greatest integer less than or equal to [tex]\( x \)[/tex]. We'll evaluate the function at each [tex]\( x \)[/tex]-value and compare it to the given [tex]\( y \)[/tex]-value of each point.
1. Point: (-5.5, -4)
- Evaluate [tex]\( f(-5.5) \)[/tex]:
[tex]\[ f(-5.5) = \lfloor -5.5 \rfloor + 2 = -6 + 2 = -4 \][/tex]
- The y-value is [tex]\(-4\)[/tex], which matches the provided point.
- Conclusion: The point (-5.5, -4) lies on the graph.
2. Point: (-3.8, -2)
- Evaluate [tex]\( f(-3.8) \)[/tex]:
[tex]\[ f(-3.8) = \lfloor -3.8 \rfloor + 2 = -4 + 2 = -2 \][/tex]
- The y-value is [tex]\(-2\)[/tex], which matches the provided point.
- Conclusion: The point (-3.8, -2) lies on the graph.
3. Point: (-1.1, 1)
- Evaluate [tex]\( f(-1.1) \)[/tex]:
[tex]\[ f(-1.1) = \lfloor -1.1 \rfloor + 2 = -2 + 2 = 0 \][/tex]
- The y-value is [tex]\(1\)[/tex], which does not match the function value.
- Conclusion: The point (-1.1, 1) does not lie on the graph.
4. Point: (-0.9, 2)
- Evaluate [tex]\( f(-0.9) \)[/tex]:
[tex]\[ f(-0.9) = \lfloor -0.9 \rfloor + 2 = -1 + 2 = 1 \][/tex]
- The y-value is [tex]\(2\)[/tex], which does not match the function value.
- Conclusion: The point (-0.9, 2) does not lie on the graph.
5. Point: (2.2, 5)
- Evaluate [tex]\( f(2.2) \)[/tex]:
[tex]\[ f(2.2) = \lfloor 2.2 \rfloor + 2 = 2 + 2 = 4 \][/tex]
- The y-value is [tex]\(5\)[/tex], which does not match the function value.
- Conclusion: The point (2.2, 5) does not lie on the graph.
6. Point: (4.7, 6)
- Evaluate [tex]\( f(4.7) \)[/tex]:
[tex]\[ f(4.7) = \lfloor 4.7 \rfloor + 2 = 4 + 2 = 6 \][/tex]
- The y-value is [tex]\(6\)[/tex], which matches the provided point.
- Conclusion: The point (4.7, 6) lies on the graph.
Therefore, the points that lie on the graph of [tex]\( f(x) = \lfloor x \rfloor + 2 \)[/tex] are:
- [tex]\((-5.5, -4)\)[/tex]
- [tex]\((-3.8, -2)\)[/tex]
- [tex]\((4.7, 6)\)[/tex]
1. Point: (-5.5, -4)
- Evaluate [tex]\( f(-5.5) \)[/tex]:
[tex]\[ f(-5.5) = \lfloor -5.5 \rfloor + 2 = -6 + 2 = -4 \][/tex]
- The y-value is [tex]\(-4\)[/tex], which matches the provided point.
- Conclusion: The point (-5.5, -4) lies on the graph.
2. Point: (-3.8, -2)
- Evaluate [tex]\( f(-3.8) \)[/tex]:
[tex]\[ f(-3.8) = \lfloor -3.8 \rfloor + 2 = -4 + 2 = -2 \][/tex]
- The y-value is [tex]\(-2\)[/tex], which matches the provided point.
- Conclusion: The point (-3.8, -2) lies on the graph.
3. Point: (-1.1, 1)
- Evaluate [tex]\( f(-1.1) \)[/tex]:
[tex]\[ f(-1.1) = \lfloor -1.1 \rfloor + 2 = -2 + 2 = 0 \][/tex]
- The y-value is [tex]\(1\)[/tex], which does not match the function value.
- Conclusion: The point (-1.1, 1) does not lie on the graph.
4. Point: (-0.9, 2)
- Evaluate [tex]\( f(-0.9) \)[/tex]:
[tex]\[ f(-0.9) = \lfloor -0.9 \rfloor + 2 = -1 + 2 = 1 \][/tex]
- The y-value is [tex]\(2\)[/tex], which does not match the function value.
- Conclusion: The point (-0.9, 2) does not lie on the graph.
5. Point: (2.2, 5)
- Evaluate [tex]\( f(2.2) \)[/tex]:
[tex]\[ f(2.2) = \lfloor 2.2 \rfloor + 2 = 2 + 2 = 4 \][/tex]
- The y-value is [tex]\(5\)[/tex], which does not match the function value.
- Conclusion: The point (2.2, 5) does not lie on the graph.
6. Point: (4.7, 6)
- Evaluate [tex]\( f(4.7) \)[/tex]:
[tex]\[ f(4.7) = \lfloor 4.7 \rfloor + 2 = 4 + 2 = 6 \][/tex]
- The y-value is [tex]\(6\)[/tex], which matches the provided point.
- Conclusion: The point (4.7, 6) lies on the graph.
Therefore, the points that lie on the graph of [tex]\( f(x) = \lfloor x \rfloor + 2 \)[/tex] are:
- [tex]\((-5.5, -4)\)[/tex]
- [tex]\((-3.8, -2)\)[/tex]
- [tex]\((4.7, 6)\)[/tex]
