The first, third, and fourth answer choices represent a function.
Step-by-step explanation:
A relation is a relationship between sets of values. The two quantities that are being related to each other are the input (x-variable) and the output (y-variable). But relations in general aren't always a good way to relate between x and y.
Say that I have situation where I want to give x dollars to the cashier so he can change them to y quarters. Here is a "example" of the relation:
Dollars (x) | Quarters (y)
0 | 0
1 | 4
2 | 8
2 | 12
Do you see something wrong here? Yes! We all know that you can't exchange 2 dollars for 12 quarters. You can only exchange 2 dollars for 8 quarters and only 8 quarters. This is a general reason why we don't rely on general relations for real-life situations. One x-variable does not exactly map to one and only one y-variable.
However, a relation that can map one x-variable to one and only one y-variable is known as a function. Let's make the above example an actual function to prove a point:
Dollars (x) | Quarters (y)
0 | 0
1 | 4
2 | 8
3 | 12
Now, the 3 dollars make 12 quarters as it should. This is how a function should look like.
There are two ways to check if a relation is a function. On a relation, table, or a set of ordered pairs, you have to make sure there is no "x-variable" that repeats. All x-values of a relation have to be unique in order to be a function. On a graph, you can also perform the Vertical Line Test. If you draw vertical lines over a relation and if the lines cross only once, then it is a function. If not, it fails the Vertical Line Test.
So to answer you're question, the first, third, and fourth choices are functions because they all have unique x-variables. The second choice is not a function because it fails the Vertical Line Test.
