To determine which of the given lists of ordered pairs represents a function, we need to remember the definition of a function: a relation is a function if every input (or x-value) is associated with exactly one output (or y-value). This means that for each x-value in the ordered pairs, there should be only one corresponding y-value.
Let's analyze each list of ordered pairs provided:
### List A: [tex]\((0,2), (4,2), (0,-4), (4,-2)\)[/tex]
- For [tex]\(x = 0\)[/tex], we have two pairs: [tex]\((0, 2)\)[/tex] and [tex]\((0, -4)\)[/tex]. Since the x-value 0 maps to two different y-values (2 and -4), this is not a function.
- For [tex]\(x = 4\)[/tex], we also have two pairs: [tex]\((4, 2)\)[/tex] and [tex]\((4, -2)\)[/tex]. This is another inconsistency.
Thus, List A is not a function.
### List B: [tex]\((2,4), (-2,4), (3,9), (-2,-4)\)[/tex]
- For [tex]\(x = 2\)[/tex], we have one pair: [tex]\((2, 4)\)[/tex].
- For [tex]\(x = -2\)[/tex], we have two pairs: [tex]\((-2, 4)\)[/tex] and [tex]\((-2, -4)\)[/tex]. Since the x-value -2 maps to two different y-values (4 and -4), this is not a function.
- For [tex]\(x = 3\)[/tex], we have one pair: [tex]\((3, 9)\)[/tex].
Thus, List B is not a function.
### List C: [tex]\((1,1), (2,3), (1,5), (4,7)\)[/tex]
- For [tex]\(x = 1\)[/tex], we have two pairs: [tex]\((1, 1)\)[/tex] and [tex]\((1, 5)\)[/tex]. Since the x-value 1 maps to two different y-values (1 and 5), this is not a function.
- For [tex]\(x = 2\)[/tex], we have one pair: [tex]\((2, 3)\)[/tex].
- For [tex]\(x = 4\)[/tex], we have one pair: [tex]\((4, 7)\)[/tex].
Thus, List C is not a function.
### List D: [tex]\((2,4), (3,9), (4,16), (5,25)\)[/tex]
- For [tex]\(x = 2\)[/tex], we have one pair: [tex]\((2, 4)\)[/tex].
- For [tex]\(x = 3\)[/tex], we have one pair: [tex]\((3, 9)\)[/tex].
- For [tex]\(x = 4\)[/tex], we have one pair: [tex]\((4, 16)\)[/tex].
- For [tex]\(x = 5\)[/tex], we have one pair: [tex]\((5, 25)\)[/tex].
Each x-value maps to a unique y-value, indicating that there are no repeated x-values with different y-values.
Thus, List D is a function.
So, the correct answer is D. [tex]\( (2, 4), (3, 9), (4, 16), (5, 25) \)[/tex].
