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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 [tex]\( x \)[/tex]-coordinate given by the points and then check if it corresponds to the [tex]\( y \)[/tex]-coordinate.
The ceiling function [tex]\(\lceil x \rceil\)[/tex] rounds [tex]\( x \)[/tex] up to the nearest integer. Let's apply this to each point and check if the resulting value matches the given [tex]\( y \)[/tex]-coordinate.
1. Point [tex]\((-5.5, -4)\)[/tex]:
- Evaluate [tex]\(\lceil -5.5 \rceil\)[/tex]: The ceiling of [tex]\(-5.5\)[/tex] is [tex]\(-5\)[/tex].
- Calculate [tex]\( f(-5.5) = \lceil -5.5 \rceil + 2 = -5 + 2 = -3 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate given is [tex]\(-4\)[/tex], which does not match [tex]\(-3\)[/tex]. So, [tex]\((-5.5, -4)\)[/tex] is not on the graph.
2. Point [tex]\((-3.8, -2)\)[/tex]:
- Evaluate [tex]\(\lceil -3.8 \rceil\)[/tex]: The ceiling of [tex]\(-3.8\)[/tex] is [tex]\(-4\)[/tex].
- Calculate [tex]\( f(-3.8) = \lceil -3.8 \rceil + 2 = -4 + 2 = -2 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate given is [tex]\(-2\)[/tex], which matches [tex]\(-2\)[/tex]. So, [tex]\((-3.8, -2)\)[/tex] is indeed on the graph.
3. Point [tex]\((-1.1, 1)\)[/tex]:
- Evaluate [tex]\(\lceil -1.1 \rceil\)[/tex]: The ceiling of [tex]\(-1.1\)[/tex] is [tex]\(-1\)[/tex].
- Calculate [tex]\( f(-1.1) = \lceil -1.1 \rceil + 2 = -1 + 2 = 1 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate given is [tex]\( 1 \)[/tex], which matches [tex]\( 1 \)[/tex]. So, [tex]\((-1.1, 1)\)[/tex] is indeed on the graph.
4. Point [tex]\((-0.9, 2)\)[/tex]:
- Evaluate [tex]\(\lceil -0.9 \rceil\)[/tex]: The ceiling of [tex]\(-0.9\)[/tex] is [tex]\( 0 \)[/tex].
- Calculate [tex]\( f(-0.9) = \lceil -0.9 \rceil + 2 = 0 + 2 = 2 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate given is [tex]\( 2 \)[/tex], which matches [tex]\( 2 \)[/tex]. So, [tex]\((-0.9, 2)\)[/tex] is indeed on the graph.
5. Point [tex]\((2.2, 5)\)[/tex]:
- Evaluate [tex]\(\lceil 2.2 \rceil\)[/tex]: The ceiling of [tex]\( 2.2 \)[/tex] is [tex]\( 3 \)[/tex].
- Calculate [tex]\( f(2.2) = \lceil 2.2 \rceil + 2 = 3 + 2 = 5 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate given is [tex]\( 5 \)[/tex], which matches [tex]\( 5 \)[/tex]. So, [tex]\((2.2, 5)\)[/tex] is indeed on the graph.
6. Point [tex]\((4.7, 6)\)[/tex]:
- Evaluate [tex]\(\lceil 4.7 \rceil\)[/tex]: The ceiling of [tex]\( 4.7 \)[/tex] is [tex]\( 5 \)[/tex].
- Calculate [tex]\( f(4.7) = \lceil 4.7 \rceil + 2 = 5 + 2 = 7 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate given is [tex]\( 6 \)[/tex], which does not match [tex]\( 7 \)[/tex]. So, [tex]\((4.7, 6)\)[/tex] is not on the graph.
In conclusion, the points that lie on the graph of the function [tex]\( f(x) = \lceil x \rceil + 2 \)[/tex] are:
- [tex]\((-1.1, 1)\)[/tex]
- [tex]\((-0.9, 2)\)[/tex]
- [tex]\((2.2, 5)\)[/tex]
The ceiling function [tex]\(\lceil x \rceil\)[/tex] rounds [tex]\( x \)[/tex] up to the nearest integer. Let's apply this to each point and check if the resulting value matches the given [tex]\( y \)[/tex]-coordinate.
1. Point [tex]\((-5.5, -4)\)[/tex]:
- Evaluate [tex]\(\lceil -5.5 \rceil\)[/tex]: The ceiling of [tex]\(-5.5\)[/tex] is [tex]\(-5\)[/tex].
- Calculate [tex]\( f(-5.5) = \lceil -5.5 \rceil + 2 = -5 + 2 = -3 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate given is [tex]\(-4\)[/tex], which does not match [tex]\(-3\)[/tex]. So, [tex]\((-5.5, -4)\)[/tex] is not on the graph.
2. Point [tex]\((-3.8, -2)\)[/tex]:
- Evaluate [tex]\(\lceil -3.8 \rceil\)[/tex]: The ceiling of [tex]\(-3.8\)[/tex] is [tex]\(-4\)[/tex].
- Calculate [tex]\( f(-3.8) = \lceil -3.8 \rceil + 2 = -4 + 2 = -2 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate given is [tex]\(-2\)[/tex], which matches [tex]\(-2\)[/tex]. So, [tex]\((-3.8, -2)\)[/tex] is indeed on the graph.
3. Point [tex]\((-1.1, 1)\)[/tex]:
- Evaluate [tex]\(\lceil -1.1 \rceil\)[/tex]: The ceiling of [tex]\(-1.1\)[/tex] is [tex]\(-1\)[/tex].
- Calculate [tex]\( f(-1.1) = \lceil -1.1 \rceil + 2 = -1 + 2 = 1 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate given is [tex]\( 1 \)[/tex], which matches [tex]\( 1 \)[/tex]. So, [tex]\((-1.1, 1)\)[/tex] is indeed on the graph.
4. Point [tex]\((-0.9, 2)\)[/tex]:
- Evaluate [tex]\(\lceil -0.9 \rceil\)[/tex]: The ceiling of [tex]\(-0.9\)[/tex] is [tex]\( 0 \)[/tex].
- Calculate [tex]\( f(-0.9) = \lceil -0.9 \rceil + 2 = 0 + 2 = 2 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate given is [tex]\( 2 \)[/tex], which matches [tex]\( 2 \)[/tex]. So, [tex]\((-0.9, 2)\)[/tex] is indeed on the graph.
5. Point [tex]\((2.2, 5)\)[/tex]:
- Evaluate [tex]\(\lceil 2.2 \rceil\)[/tex]: The ceiling of [tex]\( 2.2 \)[/tex] is [tex]\( 3 \)[/tex].
- Calculate [tex]\( f(2.2) = \lceil 2.2 \rceil + 2 = 3 + 2 = 5 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate given is [tex]\( 5 \)[/tex], which matches [tex]\( 5 \)[/tex]. So, [tex]\((2.2, 5)\)[/tex] is indeed on the graph.
6. Point [tex]\((4.7, 6)\)[/tex]:
- Evaluate [tex]\(\lceil 4.7 \rceil\)[/tex]: The ceiling of [tex]\( 4.7 \)[/tex] is [tex]\( 5 \)[/tex].
- Calculate [tex]\( f(4.7) = \lceil 4.7 \rceil + 2 = 5 + 2 = 7 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate given is [tex]\( 6 \)[/tex], which does not match [tex]\( 7 \)[/tex]. So, [tex]\((4.7, 6)\)[/tex] is not on the graph.
In conclusion, the points that lie on the graph of the function [tex]\( f(x) = \lceil x \rceil + 2 \)[/tex] are:
- [tex]\((-1.1, 1)\)[/tex]
- [tex]\((-0.9, 2)\)[/tex]
- [tex]\((2.2, 5)\)[/tex]
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