To determine which sets of pairs could be functions, we need to ensure that each set adheres to the definition of a function. In mathematical terms, a set of pairs is a function if and only if every input (x-value) is associated with exactly one output (y-value).
Let's examine each set of pairs:
Set A: [tex]\(\{(2,5),(4,-3),(-2,1),(1,6)\}\)[/tex]
- For [tex]\(x = 2\)[/tex], the output is 5.
- For [tex]\(x = 4\)[/tex], the output is -3.
- For [tex]\(x = -2\)[/tex], the output is 1.
- For [tex]\(x = 1\)[/tex], the output is 6.
Each x-value is paired with one unique y-value, so Set A could be a function.
Set B: [tex]\(\{(1,3),(2,1),(3,5),(1,-3)\}\)[/tex]
- For [tex]\(x = 1\)[/tex], the outputs are 3 and -3.
- For [tex]\(x = 2\)[/tex], the output is 1.
- For [tex]\(x = 3\)[/tex], the output is 5.
The x-value 1 is associated with two different y-values (3 and -3), so Set B cannot be a function.
Set C: [tex]\(\{(-1,1),(0,0),(1,1),(-2,2)\}\)[/tex]
- For [tex]\(x = -1\)[/tex], the output is 1.
- For [tex]\(x = 0\)[/tex], the output is 0.
- For [tex]\(x = 1\)[/tex], the output is 1.
- For [tex]\(x = -2\)[/tex], the output is 2.
Each x-value is paired with one unique y-value, so Set C could be a function.
Set D: [tex]\(\{(0,0),(2,4),(2,-4),(3,9)\}\)[/tex]
- For [tex]\(x = 0\)[/tex], the output is 0.
- For [tex]\(x = 2\)[/tex], the outputs are 4 and -4.
- For [tex]\(x = 3\)[/tex], the output is 9.
The x-value 2 is associated with two different y-values (4 and -4), so Set D cannot be a function.
Set E: [tex]\(\{(-2,8),(-3,-27),(-1,1),(3,27)\}\)[/tex]
- For [tex]\(x = -2\)[/tex], the output is 8.
- For [tex]\(x = -3\)[/tex], the output is -27.
- For [tex]\(x = -1\)[/tex], the output is 1.
- For [tex]\(x = 3\)[/tex], the output is 27.
Each x-value is paired with one unique y-value, so Set E could be a function.
Based on our analysis, the sets that could be functions are:
- Set A
- Set C
- Set E
