Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
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]
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
