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Video: Determining Whether Relations Are Functions

Alex Cutbill

Which of the relations shown by the set of ordered pairs below does not represent a function? [A] {(−8, 4), (−9, 4), (10, 4), (11, 4)} [B] {(3, 4), (11, 8), (3, 12), (11, 11)} [C] {(3, 11), (11, 19), (27, 35), (43, 43)} [D] {(3, 11), (4, 11), (5, 11), (6, 11)}

09:54

Video Transcript

Which of the relations shown by the set of ordered pairs below does not represent a function? So we have four options here, each of which is a set of ordered pairs, each of which is a relation, but only three of those four also represent a function.

Let’s go through the options one by one, starting with option A. So option A is the set of ordered pairs minus eight, four; minus nine, four; 10, four; and 11, four. And this defines a relation. Let’s first remind our self what relation is A relation is defined on two sets, which I’m going to call 𝑥 and 𝑦. And let’s see how this works by filling in the relation defined by the set an option A.

The first element of the set is the ordered pair minus eight, four. And the fact that the first component of that ordered pair is minus eight tells us that minus eight has to go in the set 𝑥, and the fact that the second component of that ordered pair is four tells us that four has to go in the set 𝑦. But more than that, we’re told not just a minus eight and four in those two sets. We’re told that there’s a relation between these two elements. We represent that on the diagram using an arrow, like so.

Okay, so let’s move on to the second element of our set, which is the ordered pair minus nine, four. The first component that ordered pair is minus nine, so that goes in 𝑥. The second component, four, you already put in our set 𝑦; we don’t need to draw it again. We just need to represent the relation using an arrow. And I’ve done the same, the last two elements of the set, which were 10, four and 11, four.

Now perhaps 𝑥 contains some other things apart from minus eight, minus nine, 10, and 11. It could be the set of integers for example. And similarly, 𝑦 might be more than just a set of the number four. It could be the set of whole numbers. In fact, both of the sets could be the set of the real numbers, but we’ve just marked those numbers, those elements of those sets which are involved in this relation here. And that’s all we need to do. This is a perfectly good relation, but the question is is it also a function.

So let’s ignore the diagram for the moment and just have a look at the set. If this is a function, then each ordered pair represents a pair of input and output values of that function. So for example, the first ordered pair — minus eight, four — tells us that minus eight is an input whose output given by the function is four Moving along, we see that the input nine has the output four as well. As does the input 10, it has the output four. And the final pair of input and output values tells us that the input 11 gives the output four.

Going back to the diagram, we can say that the set 𝑥 is now the set of inputs to that function, otherwise known as the domain of the function. And the set 𝑦 is the set of outputs of the function, otherwise known as the range of the function. And looking at the diagram, we see that every input in the domain is given an output in the range, and there’s no problem at all about that. So this is a function; A is a function. And so it’s not our answer. Let’s move on to the second option, B.

Option B is quite a similar looking set of ordered pairs to what we had before, and we’re going to go through the same process of representing this relation on a diagram. So the first element of this set is the ordered pair three, four. The first component of the pair, three, goes in the set 𝑥 the second component, four, goes in the set 𝑦. And of course, they are related. There’s a relation between them, so we draw an arrow. Next, we have 11, eight. We do the same, so 11 goes in the set 𝑥; eight goes in the set 𝑦; and we draw an arrow between them.

So far so good. How about three, 12? Well three goes in the set 𝑥. Oh! we’ve already got three in the set 𝑥; you don’t need to draw it again. We’ve got 12, the second component, has to go in the set 𝑦. And we draw an arrow between them. And finally, 11, 11. So 11 in the first component means that it’s 11 in the first set 𝑥, which we already have, and 11 as a second component of the ordered pair tells us that we need to put 11 into the set 𝑦 as well, so this is a different 11 because it’s in a different set. And we showed that these related like so.

I just like to stress this is a perfectly fine relation. In fact, any set of ordered pairs is a perfectly fine relation that you can establish like we have. But the question is is it also a function. So remember, a function can be defined as the set of ordered pairs of the input and output values of that function. So for example the first ordered pair — three, four — in that set tells us that the input three when given to that function gives the output four. So maybe if the function is called 𝑓, we’ll write 𝑓 of three is equal to four.

The next ordered pair tells us that 𝑓 of 11, the input, is eight, the output. Okay, what about the third ordered pair? Well we’ve got an input of three gives an output of 12, and so 𝑓 of three is 12. But hang on, we just said that 𝑓 of three was four. How can that make sense? How can 𝑓 of three be both four and 12? Well it can’t.

We can notice the same thing on the diagram. I’ve got three going to four and also to 12, so there’re two arrows coming out of the number three on the left. Again this is fine for a relation, but not fine for a function. We can’t have 𝑓 of three being both four and 12, so this set of ordered pairs does not define a function. As a result, B is our answer. This is the set of ordered pairs which defines a relation but it does not represent a function.

Just to conclude, let’s see a quick way of finding whether a set of ordered pairs represents a function. Look at the first components of those ordered pairs, so three, 11, three, and 11, and you’ll notice that there are some repeats there. So there’s some inputs which are given twice, for example three, with different outputs, four and 12, and 11 with eight and 11. So if you ever see repeats in the first components of the ordered pairs, you know that’s not going to be a function.

And if you have a diagram which represents relation, like we do below, you can ask whether that also represents a function by thinking about it as a mapping diagram. And as we saw before, three was mapped to both four and 12. There were two arrows coming out of three from the left going to the set of outputs as we like to think about it on the right, and of course it doesn’t make sense. So if there are two or more arrows coming out of any element on the left, then we know that’s not a function

You can go back to earlier in the video to see that these two problems didn’t occur with option A, so that was fine; that was a function. And you can check as well that options C and D do represent functions. You can check that in each of those options in each of those sets, the first component of those ordered pairs is never repeated. And you can check that when you draw a diagram from those relations, there’s never more than one arrow coming out of any input on the left.