Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.
Relations and functions are a huge part of mathematics. In this lesson, we will define relations and functions. We will look at the differences between the two and how to determine whether a given relation is a function.
Have you ever made a blanket? If you have, then you know it takes fabric to do so. Suppose you are making a blanket by sewing together swatches of fabric. You go to the store, and there is a sale on these swatches. You get three swatches for $4.00, regardless of whether you buy one, two, or three, and each swatch after that costs an additional $2.00. The amount of money you spend is related to how many swatches you buy. In mathematics, this is an example of a relation. A relation consists of two sets of elements called inputs and outputs, where the input is related to the output in some way. In our example, the cost is the input, and the number of swatches is the output. We can also represent the relation using ordered pairs as follows.
We see that in this relation, when we have an input of 4, we have an output of 1, 2, or 3. This tells us that if you spent $4, you bought 1, 2, or 3 swatches of fabric. Similarly, when we have an input of 8, we have an output of 5. This tells us that if you spent $8, you bought 5 swatches of fabric.
Now suppose you go to buy your swatches on a day that the store is not having the sale. Because there is no sale, the swatches are $2 a piece.
This makes for a relation with different inputs and outputs. This can also be represented using ordered pairs as follows.
The cost is still our input, and the number of swatches is still our output. However, this is a special type of relation. Do you notice anything different about this one from the initial example? I'll give you a hint. It has to do with the first three ordered pairs in the relation.
In the first relation, when there was a sale, the first three ordered pairs all had the same input with different outputs. In this relation, when there is no sale, each input has one and only one output. When this is the case, we call the relation a function. In mathematics, a function is a relation in which no input relates to more than one output. In our example, we would say the number of swatches you buy is a function of the cost.
As we just saw, the difference between a relation that is a function and a relation that is not a function is that a relation that is a function has inputs relating to one and only one output. When a relation is not a function, this is not the case.
So, how can we determine if a relation is a function? A good way to do this is to consider the phrase 'is a function of' as 'is determined by.' When we are dealing with a function, every output is determined by its input.
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Let's consider our swatch examples. When there wasn't a sale, we found that the relation relating to cost to number of swatches was a function. That is, the number of swatches you buy 'is a function of' the cost. Thus, the number of swatches you buy 'is determined by' the cost. So if I told you your total cost was going to be $4, you could tell me, with complete certainty, that you were buying 2 swatches of fabric. Similarly, if I told you the total cost was going to be $8, you could tell me you were going to get 4 swatches. This is because each input (or cost) has exactly one output (or number of swatches).
Now, think about the first swatch example, when there was a sale. If I told you that your cost was going to be $4, would you be able to determine how many swatches you were buying? The answer is no! You couldn't, because $4 is related to 1, 2, and 3 swatches. You have no way of determining if you were getting 1, 2, or 3 swatches, so the number of swatches 'is not determined by' the cost. Thus, the number of swatches 'is not a function of' the cost. This is because the input 4 has more than one output, namely 1, 2, and 3.
In general, we can determine whether a relation is a function by looking at its inputs and outputs. If an input has more than one output, the relation is not a function. If every input has exactly one output, then the relation is a function.
Consider this relation:
What do you think? Is this relation a function? Let's think about it. If I give you an input, can you determine the output? In other words, does each input have exactly one output? If you said yes, then you're correct!
Each input has exactly one output, and if I gave you an input, you can determine the output. Thus, this relation is a function.
A relation is a set of inputs and outputs that are related in some way. When each input in a relation has exactly one output, the relation is said to be a function. To determine if a relation is a function, we make sure that no input has more than one output. For a relation to be a function, we must be able to determine an output given an input. A good way to remember this is to think of 'is a function of' as 'is determined by.'
Relations and functions are a huge part of mathematics, so it is extremely useful to be familiar with the concept of both of them as well as being able to distinguish between the two. We are now a bit more familiar with both of these.
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