which of these relations is a function?

Explanation:

Step1: Recall function - vertical line test

A relation is a function if for every x - value there is exactly one y - value. We can use the vertical - line test on the graphs. If a vertical line intersects the graph at more than one point, the relation is not a function.

Step2: Analyze the first graph (ellipse)

A vertical line can intersect the ellipse at two points. So, it is not a function.

Step3: Analyze the second graph (circle)

A vertical line can intersect the circle at two points. So, it is not a function.

Step4: Analyze the third graph (V - shaped graph)

For every x - value, there is exactly one y - value. A vertical line will intersect the V - shaped graph at most once. So, it is a function.

Step5: Analyze the fourth graph (triangle - like graph)

A vertical line can intersect the graph at two points in some places. So, it is not a function.

Answer:

The V - shaped graph (bottom - left graph) is a function.