Question

which of these relations are functions? select all that apply.

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 first graph

For the first graph (top - left), when we imagine drawing vertical lines across the graph, each vertical line will intersect the graph at most once. So it passes the vertical line test and is a function.

Step3: Analyze the second graph (top - right)

For the second graph (top - right), if we draw a vertical line, it will intersect the graph at more than one point (since it has a "loop" in the vertical direction). So it fails the vertical line test and is not a function.

Step4: Analyze the third graph (bottom - left, the ellipse)

For the ellipse (bottom - left), a vertical line (for example, the line through the center vertically) will intersect the ellipse at two points. So it fails the vertical line test and is not a function.

Step5: Analyze the fourth graph (bottom - right)

For the fourth graph (bottom - right), a vertical line will intersect the graph at more than one point (due to the curved shape that loops back vertically). So it fails the vertical line test and is not a function.

Answer:

The top - left graph (the first relation) is a function.