Find solutions to your problems with the help of IDNLearn.com's expert community. Get the information you need from our community of experts who provide accurate and comprehensive answers to all your questions.
Sagot :
To determine whether a list of ordered pairs represents a function, we need to check the definition of a function. A function is a relation where each input (or [tex]\( x \)[/tex]-value) is paired with exactly one output (or [tex]\( y \)[/tex]-value). In other words, no two ordered pairs can have the same [tex]\( x \)[/tex]-value with different [tex]\( y \)[/tex]-values.
Let's examine each option step-by-step to find out if it satisfies this criterion.
### Option A: [tex]\( (-2,3), (1,3), (3,7), (1,4) \)[/tex]
- The first pair is [tex]\((-2, 3)\)[/tex]
- The second pair is [tex]\((1, 3)\)[/tex]
- The third pair is [tex]\((3, 7)\)[/tex]
- The fourth pair is [tex]\((1, 4)\)[/tex]
In this list, the input value [tex]\(1\)[/tex] appears twice, once with output [tex]\(3\)[/tex] and once with output [tex]\(4\)[/tex]. This means that [tex]\(1\)[/tex] maps to two different [tex]\( y \)[/tex]-values. Therefore, Option A is not a function.
### Option B: [tex]\( (1,8), (2,9), (3,10), (3,11) \)[/tex]
- The first pair is [tex]\((1, 8)\)[/tex]
- The second pair is [tex]\((2, 9)\)[/tex]
- The third pair is [tex]\((3, 10)\)[/tex]
- The fourth pair is [tex]\((3, 11)\)[/tex]
In this list, the input value [tex]\(3\)[/tex] appears twice, once with output [tex]\(10\)[/tex] and once with output [tex]\(11\)[/tex]. This means that [tex]\(3\)[/tex] maps to two different [tex]\( y \)[/tex]-values. Therefore, Option B is not a function.
### Option C: [tex]\( (-1,4), (1,7), (2,10) \)[/tex]
- The first pair is [tex]\((-1, 4)\)[/tex]
- The second pair is [tex]\((1, 7)\)[/tex]
- The third pair is [tex]\((2, 10)\)[/tex]
In this list, all the input values [tex]\(-1\)[/tex], [tex]\(1\)[/tex], and [tex]\(2\)[/tex] are unique and map to only one output each. There are no repeated [tex]\( x \)[/tex]-values with different [tex]\( y \)[/tex]-values. Therefore, Option C is a function.
### Option D: [tex]\( (3,7), (4,5), (3,8) \)[/tex]
- The first pair is [tex]\((3, 7)\)[/tex]
- The second pair is [tex]\((4, 5)\)[/tex]
- The third pair is [tex]\((3, 8)\)[/tex]
In this list, the input value [tex]\(3\)[/tex] appears twice, once with output [tex]\(7\)[/tex] and once with output [tex]\(8\)[/tex]. This means that [tex]\(3\)[/tex] maps to two different [tex]\( y \)[/tex]-values. Therefore, Option D is not a function.
### Conclusion
After examining all the options, option C: [tex]\( (-1,4), (1,7), (2,10) \)[/tex] is the only list of ordered pairs that represents a function.
Let's examine each option step-by-step to find out if it satisfies this criterion.
### Option A: [tex]\( (-2,3), (1,3), (3,7), (1,4) \)[/tex]
- The first pair is [tex]\((-2, 3)\)[/tex]
- The second pair is [tex]\((1, 3)\)[/tex]
- The third pair is [tex]\((3, 7)\)[/tex]
- The fourth pair is [tex]\((1, 4)\)[/tex]
In this list, the input value [tex]\(1\)[/tex] appears twice, once with output [tex]\(3\)[/tex] and once with output [tex]\(4\)[/tex]. This means that [tex]\(1\)[/tex] maps to two different [tex]\( y \)[/tex]-values. Therefore, Option A is not a function.
### Option B: [tex]\( (1,8), (2,9), (3,10), (3,11) \)[/tex]
- The first pair is [tex]\((1, 8)\)[/tex]
- The second pair is [tex]\((2, 9)\)[/tex]
- The third pair is [tex]\((3, 10)\)[/tex]
- The fourth pair is [tex]\((3, 11)\)[/tex]
In this list, the input value [tex]\(3\)[/tex] appears twice, once with output [tex]\(10\)[/tex] and once with output [tex]\(11\)[/tex]. This means that [tex]\(3\)[/tex] maps to two different [tex]\( y \)[/tex]-values. Therefore, Option B is not a function.
### Option C: [tex]\( (-1,4), (1,7), (2,10) \)[/tex]
- The first pair is [tex]\((-1, 4)\)[/tex]
- The second pair is [tex]\((1, 7)\)[/tex]
- The third pair is [tex]\((2, 10)\)[/tex]
In this list, all the input values [tex]\(-1\)[/tex], [tex]\(1\)[/tex], and [tex]\(2\)[/tex] are unique and map to only one output each. There are no repeated [tex]\( x \)[/tex]-values with different [tex]\( y \)[/tex]-values. Therefore, Option C is a function.
### Option D: [tex]\( (3,7), (4,5), (3,8) \)[/tex]
- The first pair is [tex]\((3, 7)\)[/tex]
- The second pair is [tex]\((4, 5)\)[/tex]
- The third pair is [tex]\((3, 8)\)[/tex]
In this list, the input value [tex]\(3\)[/tex] appears twice, once with output [tex]\(7\)[/tex] and once with output [tex]\(8\)[/tex]. This means that [tex]\(3\)[/tex] maps to two different [tex]\( y \)[/tex]-values. Therefore, Option D is not a function.
### Conclusion
After examining all the options, option C: [tex]\( (-1,4), (1,7), (2,10) \)[/tex] is the only list of ordered pairs that represents a function.
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Accurate answers are just a click away at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.
