Question

Which relation is a function?



Responses

Coordinate grid with graph of a vertical line at x equals 3.
Image with alt text: Coordinate grid with graph of a vertical line at x equals 3.

A coordinate grid containing a U shape with arrows on both ends that opens to the right. The bottom portion of the U passes through the origin.
Image with alt text: A coordinate grid containing a U shape with arrows on both ends that opens to the right. The bottom portion of the U passes through the origin.

A coordinate grid with graph of a curve that increases through the third quadrant through the origin and exits the first quadrant
Image with alt text: A coordinate grid with graph of a curve that increases through the third quadrant through the origin and exits the first quadrant

A coordinate grid with the graph of a circle centered at the origin and passing through the point begin ordered pair 2 comma 1 end ordered pair.

Answer 1

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).

- **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.

- **U shape that opens to the right**: This is a function because for every x-value, there is only one corresponding y-value.

- **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.

- **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.