Bot GPT 4-o mini To determine which relations are functions, we can use the vertical line test. A relation is a function if every input (x-value) has exactly one output (y-value).
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Vertical line at x = 3: This is not a function because it has multiple points where it intersects the vertical line. For any given x-value, there can be multiple corresponding y-values.
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U shape that opens to the right: This is a function because for every x-value, there is only one corresponding y-value.
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Curve that increases through the third quadrant and exits the first quadrant: This is also a function, as it passes the vertical line test; every x-value corresponds to only one y-value.
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Circle centered at the origin: This is not a function because a vertical line can intersect the circle at two points for some x-values, meaning those x-values have multiple y-values.
Therefore, the relations that are functions are:
- The U shape opening to the right.
- The curve that increases through the third quadrant.
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