Question

which of these relations is a function?

Explanation:

Step1: Recall the vertical line test

A relation is a function if every vertical line intersects its graph at most once.

Step2: Analyze the top - left graph

For the top - left graph (the "S" - shaped curve), if we draw a vertical line, it will intersect the graph at more than one point. So it is not a function.

Step3: Analyze the top - right graph

The top - right graph is a circle. A vertical line (for example, the line \(x = 0\)) will intersect the circle at two points. So it is not a function.

Step4: Analyze the bottom - left graph

The bottom - left graph has a curve that, when a vertical line is drawn, will intersect the graph at more than one point (it has a "loop" that will be intersected twice by a vertical line). So it is not a function.

Step5: Analyze the bottom - right graph

For the bottom - right graph, if we draw any vertical line, it will intersect the graph at most once. So it passes the vertical line test.

Answer:

The graph at the bottom - right (the curve that has a peak, a valley, and crosses the \(y\) - axis) is the function.