To determine whether a relation represents a function, we must check that for each input \( x \), there is exactly one output \( y \). This means that no two ordered pairs can have the same \( x \)-value with different \( y \)-values.
Let's analyze each option:
A.
x: 3 4 5 6
y: -1 0 3 4
- Each \( x \)-value (3, 4, 5, 6) maps to exactly one \( y \)-value. This is a function.
B.
x: -4 -1 0 5
y: -7 3 -2 -7
- Each \( x \)-value (-4, -1, 0, 5) maps to exactly one \( y \)-value (although -7 is repeated for \( x = -4 \) and \( x = 5 \), it does not violate the function definition). This is a function.
C. (points (-1,0), (0,-1), (2,-2), (3,3), (5,5),(5,-5))
- The point (5, -5) shares the same \( x \)-value (5) as (5, 5) but has a different \( y \)-value. This is not a function.
D.
{(-4,3), (-2,3), (0,2),(5,2)}
- Each \( x \)-value (-4, -2, 0, 5) maps to exactly one \( y \)-value (although -2 and 0 map to the same \( y \)-value (2), this still conforms to the function definition). This is a function.
E.
x: 0 4 9
y: 2 3 4, 5
- The \( x \)-value 9 maps to two \( y \)-values (4 and 5). This is not a function.
F. (downward U shape) Points: (1,4), (0,3), (-1,0), (3,0), (-2,-5), (4,-5)
- The points (3, 0) and (-2, -5) do not share any \( x \)-values with different \( y \)-values, indicating that no input has multiple outputs. However, we don't necessarily know if this relation might have overlapping \( x \)-values in the curve itself without seeing the graph. But based on the points provided, it can be classified as a function (no repeated \( x \)-values).
Conclusion:
The relations that represent functions are:
- A
- B
- D
- F (based on the provided points)
So, the relations that represent functions are: A, B, D, and F.
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