Answer :
To determine which points lie on the graph of the function [tex]\(f(x) = \lceil x \rceil + 2\)[/tex], we need to evaluate the function at each given [tex]\(x\)[/tex]-value and see if the result matches the corresponding [tex]\(y\)[/tex]-value.
Let's analyze each point step-by-step:
1. Point [tex]\((-5.5, -4)\)[/tex]:
- [tex]\(x = -5.5\)[/tex]
- Calculate [tex]\(f(-5.5)\)[/tex]:
[tex]\[ f(-5.5) = \lceil-5.5\rceil + 2 = -5 + 2 = -3 \][/tex]
- The [tex]\(y\)[/tex]-value given is [tex]\(-4\)[/tex], which does not match [tex]\(-3\)[/tex].
- Therefore, [tex]\((-5.5, -4)\)[/tex] does not lie on the graph of [tex]\(f(x)\)[/tex].
2. Point [tex]\((-3.8, -2)\)[/tex]:
- [tex]\(x = -3.8\)[/tex]
- Calculate [tex]\(f(-3.8)\)[/tex]:
[tex]\[ f(-3.8) = \lceil-3.8\rceil + 2 = -3 + 2 = -1 \][/tex]
- The [tex]\(y\)[/tex]-value given is [tex]\(-2\)[/tex], which does not match [tex]\(-1\)[/tex].
- Therefore, [tex]\((-3.8, -2)\)[/tex] does not lie on the graph of [tex]\(f(x)\)[/tex].
3. Point [tex]\((-1.1, 1)\)[/tex]:
- [tex]\(x = -1.1\)[/tex]
- Calculate [tex]\(f(-1.1)\)[/tex]:
[tex]\[ f(-1.1) = \lceil-1.1\rceil + 2 = -1 + 2 = 1 \][/tex]
- The [tex]\(y\)[/tex]-value given is [tex]\(1\)[/tex], which matches [tex]\(1\)[/tex].
- Therefore, [tex]\((-1.1, 1)\)[/tex] lies on the graph of [tex]\(f(x)\)[/tex].
4. Point [tex]\((-0.9, 2)\)[/tex]:
- [tex]\(x = -0.9\)[/tex]
- Calculate [tex]\(f(-0.9)\)[/tex]:
[tex]\[ f(-0.9) = \lceil-0.9\rceil + 2 = -0 + 2 = 2 \][/tex]
- The [tex]\(y\)[/tex]-value given is [tex]\(2\)[/tex], which matches [tex]\(2\)[/tex].
- Therefore, [tex]\((-0.9, 2)\)[/tex] lies on the graph of [tex]\(f(x)\)[/tex].
5. Point [tex]\((2.2, 5)\)[/tex]:
- [tex]\(x = 2.2\)[/tex]
- Calculate [tex]\(f(2.2)\)[/tex]:
[tex]\[ f(2.2) = \lceil2.2\rceil + 2 = 3 + 2 = 5 \][/tex]
- The [tex]\(y\)[/tex]-value given is [tex]\(5\)[/tex], which matches [tex]\(5\)[/tex].
- Therefore, [tex]\((2.2, 5)\)[/tex] lies on the graph of [tex]\(f(x)\)[/tex].
6. Point [tex]\((4.7, 6)\)[/tex]:
- [tex]\(x = 4.7\)[/tex]
- Calculate [tex]\(f(4.7)\)[/tex]:
[tex]\[ f(4.7) = \lceil4.7\rceil + 2 = 5 + 2 = 7 \][/tex]
- The [tex]\(y\)[/tex]-value given is [tex]\(6\)[/tex], which does not match [tex]\(7\)[/tex].
- Therefore, [tex]\((4.7, 6)\)[/tex] does not lie on the graph of [tex]\(f(x)\)[/tex].
In conclusion, the points that lie on the graph of the function [tex]\(f(x) = \lceil x\rceil + 2\)[/tex] are:
[tex]\[ (-1.1, 1), (-0.9, 2), (2.2, 5). \][/tex]
Let's analyze each point step-by-step:
1. Point [tex]\((-5.5, -4)\)[/tex]:
- [tex]\(x = -5.5\)[/tex]
- Calculate [tex]\(f(-5.5)\)[/tex]:
[tex]\[ f(-5.5) = \lceil-5.5\rceil + 2 = -5 + 2 = -3 \][/tex]
- The [tex]\(y\)[/tex]-value given is [tex]\(-4\)[/tex], which does not match [tex]\(-3\)[/tex].
- Therefore, [tex]\((-5.5, -4)\)[/tex] does not lie on the graph of [tex]\(f(x)\)[/tex].
2. Point [tex]\((-3.8, -2)\)[/tex]:
- [tex]\(x = -3.8\)[/tex]
- Calculate [tex]\(f(-3.8)\)[/tex]:
[tex]\[ f(-3.8) = \lceil-3.8\rceil + 2 = -3 + 2 = -1 \][/tex]
- The [tex]\(y\)[/tex]-value given is [tex]\(-2\)[/tex], which does not match [tex]\(-1\)[/tex].
- Therefore, [tex]\((-3.8, -2)\)[/tex] does not lie on the graph of [tex]\(f(x)\)[/tex].
3. Point [tex]\((-1.1, 1)\)[/tex]:
- [tex]\(x = -1.1\)[/tex]
- Calculate [tex]\(f(-1.1)\)[/tex]:
[tex]\[ f(-1.1) = \lceil-1.1\rceil + 2 = -1 + 2 = 1 \][/tex]
- The [tex]\(y\)[/tex]-value given is [tex]\(1\)[/tex], which matches [tex]\(1\)[/tex].
- Therefore, [tex]\((-1.1, 1)\)[/tex] lies on the graph of [tex]\(f(x)\)[/tex].
4. Point [tex]\((-0.9, 2)\)[/tex]:
- [tex]\(x = -0.9\)[/tex]
- Calculate [tex]\(f(-0.9)\)[/tex]:
[tex]\[ f(-0.9) = \lceil-0.9\rceil + 2 = -0 + 2 = 2 \][/tex]
- The [tex]\(y\)[/tex]-value given is [tex]\(2\)[/tex], which matches [tex]\(2\)[/tex].
- Therefore, [tex]\((-0.9, 2)\)[/tex] lies on the graph of [tex]\(f(x)\)[/tex].
5. Point [tex]\((2.2, 5)\)[/tex]:
- [tex]\(x = 2.2\)[/tex]
- Calculate [tex]\(f(2.2)\)[/tex]:
[tex]\[ f(2.2) = \lceil2.2\rceil + 2 = 3 + 2 = 5 \][/tex]
- The [tex]\(y\)[/tex]-value given is [tex]\(5\)[/tex], which matches [tex]\(5\)[/tex].
- Therefore, [tex]\((2.2, 5)\)[/tex] lies on the graph of [tex]\(f(x)\)[/tex].
6. Point [tex]\((4.7, 6)\)[/tex]:
- [tex]\(x = 4.7\)[/tex]
- Calculate [tex]\(f(4.7)\)[/tex]:
[tex]\[ f(4.7) = \lceil4.7\rceil + 2 = 5 + 2 = 7 \][/tex]
- The [tex]\(y\)[/tex]-value given is [tex]\(6\)[/tex], which does not match [tex]\(7\)[/tex].
- Therefore, [tex]\((4.7, 6)\)[/tex] does not lie on the graph of [tex]\(f(x)\)[/tex].
In conclusion, the points that lie on the graph of the function [tex]\(f(x) = \lceil x\rceil + 2\)[/tex] are:
[tex]\[ (-1.1, 1), (-0.9, 2), (2.2, 5). \][/tex]
