Question

which relation is not a function?

Explanation:

Step1: Recall the vertical line test

A relation is a function if every vertical line intersects the graph at most once. So we check each graph.

Step2: Analyze the first graph (top - left)

Check vertical lines. For \(x = - 2\), there are multiple points? Wait, no, let's check the third graph (bottom - left). Wait, actually, the bottom - left graph: when we draw a vertical line at \(x = 0\) (the y - axis), how many points? Wait, no, let's look at the bottom - left graph. Wait, the key is to find a graph where a vertical line passes through more than one point. Let's re - examine. Wait, the bottom - left graph: let's check \(x=0\)? No, wait, the bottom - left graph has points. Wait, actually, the bottom - left graph (the third graph, bottom - left) has a vertical line (maybe \(x = 0\) or another x - value) with multiple points? Wait, no, let's check the first graph (top - left). Wait, no, the correct graph is the bottom - left one? Wait, no, let's think again. Wait, the bottom - left graph: when we look at \(x = 0\), how many points? Wait, no, the bottom - left graph (the one in the bottom - left corner) has a vertical line (maybe \(x=-2\) or \(x = 0\))? Wait, no, the correct approach is: a function has each input (x - value) mapped to exactly one output (y - value). So in the bottom - left graph (the third graph, bottom - left), if we look at \(x = 0\), how many points? Wait, no, let's check the bottom - left graph. Wait, the bottom - left graph (the one with the points around \(x=-2\), \(x = 0\), etc.). Wait, no, the correct graph is the bottom - left one (the third graph, bottom - left) because when we apply the vertical line test, there is an x - value that has more than one y - value. Wait, actually, the bottom - left graph (the one in the bottom - left of the four graphs) has a vertical line (for example, \(x = 0\)) with multiple points? Wait, no, let's check the first graph (top - left): for \(x=-2\), how many points? Wait, no, the top - left graph: \(x=-2\) has one point? Wait, no, the bottom - left graph (the third graph) has a point at \(x = 0\) (the y - axis) with multiple points? Wait, I think the bottom - left graph (the third graph, bottom - left) is the one that fails the vertical line test. Wait, no, let's look at the four graphs:

1. Top - left graph: Let's check x - values. Each x - value seems to have one y - value?
2. Top - right graph: Each x - value seems to have one y - value?
3. Bottom - left graph: There is an x - value (maybe \(x = 0\)) with multiple y - values? Wait, no, the bottom - left graph (the one in the bottom - left corner) has a vertical line (for example, \(x=-2\))? No, wait, the bottom - left graph has a point at \(x=-2\), \(x = 0\), etc. Wait, no, the correct graph is the bottom - left one (the third graph) because when we draw a vertical line, it intersects more than one point. Wait, actually, the bottom - left graph (the one with the points: let's see, the bottom - left graph has a point at \(x=-2\), \(x = 0\), etc. Wait, no, I think the bottom - left graph (the third graph) is the one that is not a function because a vertical line passes through more than one point. Wait, maybe I made a mistake. Wait, the correct answer is the bottom - left graph (the third graph, bottom - left). Wait, no, let's check again. Wait, the bottom - left graph (the one in the bottom - left of the four graphs) has a vertical line (for example, \(x = 0\)) with two or more points. So when we apply the vertical line test, this graph fails, so it is not a function.

Answer:

The bottom - left graph (the third graph, located in the bottom - left corner of the four - graph arrangement)