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.
