Question

which graph represents a function?

Explanation:

Step1: Recall the vertical line test

A relation is a function if each input (x - value) has exactly one output (y - value). So we check each graph for repeated x - values with different y - values.

Step2: Analyze the first graph

In the first graph, we check the x - coordinates of each point. Let's assume the x - coordinates (from left to right) are \(x_1,x_2,x_3,x_4,x_5,x_6\). Each x - value has only one y - value. For example, the first point has a unique x, the second point has a unique x, etc. So no two points share the same x - coordinate.

Step3: Analyze the second graph

In the second graph, the x - coordinate of the first two points (on the y - axis and the next one) - wait, looking at the columns, we can see that some x - values (columns) have multiple points. For example, the first column (x = 0) has one point, the second column (x = 1) has two points. So this fails the vertical line test.

Step4: Analyze the third graph

In the third graph, looking at the columns (x - values), some columns have multiple points. For example, there are columns with more than one point, so it fails the vertical line test.

Step5: Analyze the fourth graph

In the fourth graph, all the points are in the same column (same x - value) with different y - values. So this is a vertical line of points, which means one x - value has multiple y - values, so it fails the vertical line test.

Answer:

The first graph (the left - most graph)