Question

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

Explanation:

To determine if a relation is a function, we use the vertical line test: a graph represents a function if no vertical line intersects the graph more than once.

Step 1: Analyze the first graph (top - left)

The graph is a continuous curve. For any vertical line, it will intersect the curve at most once. So, this relation is a function.

Step 2: Analyze the second graph (top - right)

We check the vertical line test. Each x - value (domain element) has at most one y - value (range element). Even with the open and closed circles, no vertical line intersects the graph more than once. So, this relation is a function.

Step 3: Analyze the third graph (bottom - left)

We apply the vertical line test. Looking at the horizontal segments, we can see that there are x - values that would have multiple y - values (for example, some x - coordinates are covered by more than one horizontal segment). So, this relation is not a function.

Step 4: Analyze the fourth graph (bottom - right)

The graph is a continuous curve. For any vertical line, it will intersect the curve at most once. So, this relation is a function.

Answer:

The top - left, top - right, and bottom - right graphs represent functions.