Which of these sets of points define a function (points below)?

A relation is only a function when there is only one output for every input. In other words, you can't have the same #x# for different #y# values. You can however have the same #y# value for different #x# values

For a, #-2# is repeated for every #y# value. This cannot be so this relation is not a function.

For b, each #x# value has a only one unique value so this is a function.

For c, this is not a function because #-1# is used more than once

For d, This is also not a function since #-3# is used three times and #1# is used three times.

Essentially, check to see if there are any repeating #x# values. If there are, then it is not a function. If no #x# values repeat, then it is a function.

Another way to view this problem is to graph the points and use the "vertical line test".

The vertical line test is a visual way to see if, for any #x# value, there are more than 1 #y# values. If the vertical line intersects more than one point, then the equation isn't a function.

For instance, for the set of points a. {(-2, -2),(-2, -1),(-2, 0),(-2, 1),(-2, 2)}:

graph{((x+2)^2+(y+2)^2-.1)((x+2)^2+(y+1)^2-.1)((x+2)^2+(y+0)^2-.1)((x+2)^2+(y-1)^2-.1)((x+2)^2+(y-2)^2-.1)(x-0y+2)=0}

I've drawn a vertical line through more than 1 point and so the relation that created this set of points is not a function.

Contrast that with set b. {(1, 0),(-1, 0),(2, 1),(-2, 1),(3, 2),(-3, 2)}

graph{((x-1)^2+(y+0)^2-.1)((x+1)^2+(y+0)^2-.1)((x-2)^2+(y-1)^2-.1)((x+2)^2+(y-1)^2-.1)((x-3)^2+(y-2)^2-.1)((x+3)^2+(y-2)^2-.1)(x-0y+2)(x-0y+1)(x-0y+3)(x-0y-2)(x-0y-1)(x-0y-3)=0}

Each of vertical lines goes through only 1 point and so the relation that created this set of points is a function.