# What is a Function?

## Definition of a Function

A **function** is a relation where each input relates—and here’s the key—to exactly one output. The input is generally considered our x-value, but we don’t always use x, that’s why we call it an input. The output would generally be considered the y-value. Here we’re going to do two examples where we’re going to determine whether each relation is, in fact, a function.

This is a **relation** because all a relation is, is a set of ordered pairs. Each one of these is a relation, but now we need to determine if they’re a function. There are two methods for doing this. First is the mapping diagram, and then there’s the vertical line test. I’m going to show you both, starting with the mapping diagram.

I’m going to make a mapping diagram, by making two columns: one for my inputs, and one for my outputs. Your inputs are your first coordinate in each ordered pair, so our inputs here are negative 3, negative 2, negative 1, 0, and 1. We like to list things in math in order from least to greatest, and here make sure you don’t repeat any of the numbers.

In this case we didn’t have any numbers that repeated, but if we had we would have only written that number once, because really what you’re writing here is your domain, so in fact you could change the title of this to domain and range. All the domain is, is your set of inputs; and all a range is, is your set outputs.

When you’re listing the set, you only need to list each number one time. The range is your set of outputs, or the second value, so we have 2, and another 2, but we’re only going to write it once, negative 4, negative 3, and 5. Again, I’m going to list them in order from least to greatest, so negative 4, negative 3, 2, and 5.

Here’s where the **mapping** comes in, this is the really important part—it’s the visual that helps you see if your relation is a function. The negative 3 maps to 2, so that just means draw an arrow to 2. Negative 2 maps to 2, so draw another arrow to 2. Negative 1 maps to negative 4, 0 maps to negative 3, and 1 maps to 5.

Now here’s where you check to see if it’s a function. You want to look at each input value, and make sure that each input value only mapped to one output value, so negative 3 is only mapped to one number, 2. Negative 2 is only mapped to one number, 2. Negative 1 is only mapped to one number, negative 4. 0 is only mapped to negative 3, and 1 is only mapped to 5, which means this is a function.

Now let’s use the **vertical line test** to verify that it is a function. To use the vertical line test, the first thing you need to do is graph your ordered pairs. The first ordered pair is (negative 3, 2), so go over to negative 3 and up to 2. Then (negative 2, 2). Then (negative 1, negative 4), (0, negative 3), and (1, 5).

The vertical line test is just like it sounds, you’re going to take something vertically, like a pencil or a pen, (or I’m going to use my marker) and you’re going to run it along your graph making sure than in every vertical line you only see one point, and we see that here. Each vertical line only has one point on it, and that means that it is a function—passes the vertical line test, so it is a function.

Let’s look at one more example, first using the mapping diagram and then using vertical line test. Again, we’re going to make a mapping diagram listing our domain and then our range. Your domain is your set of x-values or inputs, and we generally list them from least to greatest, and we do not repeat any numbers.

Our domain would be negative 2, we have another negative 2, 0, 1, and then 3. Again, we only write that negative 2 once because we just need to know what numbers are represented in our inputs, so we have each number represented here. Your range is your set of outputs, again, in order from least to greatest.

We have 1, negative 3, 2, 0, and negative 1, so we would start with negative 3, then negative 1, then 0, 1, 2, and now we map. Negative 2 maps to a positive 1. Negative 2 maps to negative 3. 0 maps to 2, 1 maps to 0, and 3 maps to negative 1. Again, a function is where each input relates to exactly one output.

Here we have an input, negative 2, mapping to two outputs, which means this is not a function because of that one number, that one input, that maps to two different outputs, it’s not a function. Let’s look at that using the vertical line test. (Negative 2, 1), (negative 2, negative 3), (0, 2), (1, 0), and (3, negative 1).

Now just take your pencil, pen, or in my case my marker, and check to see if it passes the vertical line test by making sure that there’s only one point in each vertical line. You can see that right here in the beginning, this vertical line, fails the vertical line test because we have two points on this vertical line, which was the same input value in our mapping diagram, the negative 2. You can see, using the mapping diagram or the vertical line test, that this second relation is not a function.