When you're working with functions, mapping diagrams are a great way to see and track the ways your inputs are related to your outputs. In this lesson, we will learn how to use and draw a mapping diagram.
What Are Mapping Diagrams?
Have you ever watched a waitress as she takes orders for drinks from a large table? How does she keep track of which drink goes with which customer? Of course, sometimes they don't, but sometimes it's amazing how they track every drink at the table without looking at a diagram or anything.
Imagine our poor waitress has 12 people at a large table, and she wants to keep track of all the drink orders. She could use a mapping diagram. A mapping diagram helps you to remember relationships between one set of values and another set, or how they're paired together. For example, let's see if we can give our waitress a little help:
She numbers the customers in a clockwise order, lists the drinks as they show up in the orders, and then draws arrows between the customer numbers and the drink name. She has created a mapping diagram.
Mapping diagram for the waitress
Using Mapping Diagrams
Mapping diagrams are useful when we're working with functions. They allow us to track the relationship between the inputs (the numbers you're putting into the 'machine') and the outputs (the numbers that are coming out). We can use the diagram to show which input values are tracked to which output values. They also help us make sure a function really is a function.
For example, say we have the following set of pairs, where the first number is the input and the second number is the output.
Mapping diagram for input and output values
Creating a Mapping Diagram
When we construct a mapping diagram from the list of input and output values, we'll draw an area for the inputs and an area for the outputs. We'll list the inputs and outputs in their own areas and then draw arrows to show which input value leads to which output value.
Notice each number from the inputs is tied to only one of the outputs. In math, that's the test for a function. Remember, a function must assign only one output value for each possible input. If any input value has more than one output, then you may have a relationship, but it's not technically a function. Let's look at one that's like that:
This is not a function!
Can we create a mapping diagram from a function that is an equation? Absolutely! All you have to do is decide what input values you want to work with and then plug them into the equation. Your output values will be whatever you get from the equation.
For example, say you have the function f that is described as f(x) = 5x - 3. Just pick some values you want to plug into the formula. For this example, we'll use -3, -2, -1, 0, 1, 2, and 3. Plugging them into our equation, we have the following output values:
5(-3) - 3 = -18
5(-2) - 3 = -13
5(-1) - 3 = -8
5(0) - 3 = -3
5(1) - 3 = 2
5(2) - 3 = 7
5(3) - 3 = 12
Now we can create our mapping diagram. Remember, set up an area for your input values (-3, -2, -1, 0, 1, 2, 3) and an area for your output values (-18, -13, -8, -3, 2, 7, 12). Now show the connections between the inputs and the outputs. You can use ovals, boxes, and any other shape you like, but make sure your values line up so it's easy to connect them with arrows.
Mapping your f(x) = 5x - 3 function
Mapping Diagrams and Graphs
So, what if you had a graph instead? Could you map the points on the graph? Absolutely! For example, say we have a graph of the number of songs downloaded by a friend in 2015.
Downloaded songs graph
If we take a look at the graph, we can see just how many songs we downloaded in each of the months of 2015. It jumped around quite a bit. So let's map those graph positions to a mapping diagram. We'll use the months for the input and the number of songs downloaded for the output.
Mapping diagram for our downloaded songs graph
A mapping diagram is used to show the relationship or the pairing up between a set of inputs and a set of outputs. You can use it to check whether a relationship is a function by making sure that no single input is linked to more than one output. You can construct mapping diagrams from real-life situations, equations, related sets of points, and graphs. Mapping diagrams are a good way to visualize how two sets of values are related.