Question

which of these relations is a function?

Explanation:

Step1: Recall vertical - line test

A relation is a function if for every \(x\) - value there is exactly one \(y\) - value. The vertical - line test is used to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the relation is not a function.

Step2: Analyze first graph

In the first graph, a vertical line can be drawn that intersects the curve at more than one point. So, it is not a function.

Step3: Analyze second graph

In the second graph, a vertical line can be drawn (the dashed vertical asymptote region shows that for some \(x\) - values there are multiple \(y\) - values approaching from different directions). So, it is not a function.

Step4: Analyze third graph

For the third graph, any vertical line drawn will intersect the curve at exactly one point. So, it is a function.

Step5: Analyze fourth graph

In the fourth graph (a circle), a vertical line can be drawn that intersects the circle at two points. So, it is not a function.

Answer:

The third graph is a function.