Question

which of these relations is a function?

Explanation:

Step1: Recall the definition of a function

A relation is a function if every input (x - value) has exactly one output (y - value). We can use the vertical line test: if any vertical line drawn through the graph intersects the graph at more than one point, the relation is not a function.

Step2: Analyze the first graph (top - left)

Looking at the top - left graph, if we draw a vertical line, it will intersect the graph at more than one point (since the graph has a "zig - zag" shape). So this relation is not a function.

Step3: Analyze the second graph (top - right)

In the top - right graph, when we consider the x - values of the points, there are x - values that are associated with more than one y - value (for example, if we look at the horizontal and the other lines, some x - values have multiple y - values). So this relation is not a function.

Step4: Analyze the third graph (bottom - left)

For the bottom - left graph, if we apply the vertical line test, any vertical line will intersect the graph at most once. Each x - value has exactly one y - value.

Step5: Analyze the fourth graph (bottom - right)

In the bottom - right graph, a vertical line will intersect the graph at more than one point (since it is a curve that opens sideways), so this relation is not a function.

Answer:

The relation represented by the bottom - left graph (the one with the blue curve that has a peak and a valley and the separate blue line with a dot) is a function.