Which of these relations are functions

The fourth and fifth options.

Step-by-step explanation:

Now, there are two key points to this question that are essential - first, in a relation that is a function, for every input, there is only one single output. For instance, in the first option, we can see that when x=2, y=11 and 15. Hence, it is ruled out. The same goes for the third option; of the coordinates, it is clear that certain values overlap, and in the final option, when x= -4, y= -2 and -5.

In the fifth option, however, the input (x) leads to only one output value (y) each. Therefore, it is a function.

Secondly,  the vertical line test is a visual way to determine if a curve is a graph of a function or not. The best way to do this is to either visualise a vertical straight line, or get a ruler of some sort, and drag it across the graph. If at any time, the vertical line touches more than one point, it is not a function.

If the coordinates in the second option are joined, we can see that two of them have the same x-coordinate - thus, the line would touch two points, and it is not a function.

In the fourth option, on the other hand, the vertical line only touches one point throughout the entire graph - therefore, it is a function.

Hope this helps!

include a picture of the drawing in order for me to

step-by-step explanation: