Question

which of the following relations is not a function?
(x) (y)
-1 4
1 5
-1 6
2 7
-2 8
2 9
{(4,1),(3,-5),(2,1),(-4,1),(-2,5)}
(x) (y)
0 2
1 2
2 2
5 2
8 2
12 2

Explanation:

Step1: Recall the definition of a function

A relation is a function if each input (x - value) has exactly one output (y - value). So we need to check each relation to see if any x - value is paired with more than one y - value.

Step2: Analyze the first table (left - top)

The x - values are - 1, 1, - 1, 2, - 2, 2.

• For \(x=-1\), we have \(y = 4\) and \(y = 6\) (same x - value with different y - values).
• For \(x = 2\), we have \(y=7\) and \(y = 9\) (same x - value with different y - values). So this relation is not a function. But let's check the others to be sure.

Step3: Analyze the set of ordered pairs \(\{(4,1),(3, - 5),(2,1),(-4,1),(-2,5)\}\)

The x - values are 4, 3, 2, - 4, - 2. Each x - value appears only once, so each has exactly one y - value. This is a function.

Step4: Analyze the graph (right - top)

To check the graph, we can use the vertical line test. A vertical line should intersect the graph at most once. Looking at the points, each x - value (the x - coordinate of each point) is unique. So no vertical line will intersect more than one point, so it is a function.

Step5: Analyze the last table (right - bottom)

The x - values are 0, 1, 2, 5, 8, 12. Each x - value appears only once, and they all map to \(y = 2\). Since each x has exactly one y (even though the y - value is the same for all x), this is a function.

Answer:

The relation in the left - top table (with x - values - 1, 1, - 1, 2, - 2, 2 and corresponding y - values 4, 5, 6, 7, 8, 9) is NOT a function.