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We examine some relations using mapping diagrams and learn about domains (sets of input values) and ranges (sets of output values). We also look at Cartesian graphs to see how they relate to their mapping diagrams and discuss their domains and ranges.
Mapping Diagrams: Domain and Range
This is a mapping diagram.
So what a mapping diagram means is if we have every value in set A, we can transform it to give us every value in set B.
Now in this case if we take two and we add four, we get six; if we take four and we add four, we get eight.
So then the last one, we have 𝑥 and we add four, we get 𝑥 and four.
Now 𝑥 add four: so it’s obviously it would be a graph if we said 𝑦 equals 𝑥 add four.
So I’ll show you how Cartesian graph is actually exactly the same as one of these mapping diagrams.
So this is the graph of 𝑦 equals 𝑥 add four. Now let’s look at our input values.
So, in our mapping diagram, if we input two, then we receive six.
So, in the graph, if we input two, then you receive six.
And in our mapping diagram, if we input four, we receive eight.
And then in our graph, if we input four, we receive eight.
Now these input and these output values have two very special names.
So the input values are called “the domain.” So think about that domain. And our domain will always represent the 𝑥-axis. And then the output values are called “the range.” And the range will always represent the 𝑦-axis.
Or if we take it back to our mapping diagram, we’re saying the domain are our input values or the values in set A.
And then the range are our output values or the values in set B.
Now this is relatively simple: putting one value in, you get one value out.
So let’s look at another example.
Okay taking a second to look at this example, we can see the thing that’s different here is that there are two input values giving us just one output value.
So we would call this “a many-to-one function.”
Now in this case if we have minus one, what do we have to do to it to get one?
We would have to times it by minus one.
So this is saying everything on the left-hand side, all of the input values, we will have to square to get the output values.
So 𝑥 will then become 𝑥 squared.
And again the input, the domain, and then the output will be the range.
Now let’s have a look at this Cartesian graph as well to really understand it.
So this is our Cartesian graph.
And we can see how it relates to the mapping diagram.
So if we input minus one in the mapping diagram, we get one.
And same if we input one, we also get one. So in the graph if we input minus one, we get one. And same if we input one, we also get one.
Now looking at minus two: if we input minus two, we get four.
And also if we input two, we get four.
So that tells us that two values in the domain give us one value in the range.
And again domain is the values you put in — domain the 𝑥-axis — and the range will be the 𝑦-axis.
So in summary, this is a mapping diagram where we input some value into set A and it gives us some value in set B.
And we always have to transform it in the same way.
So in this case, what we’ve done is we transformed set A by adding two each time.
So then 𝑥 would be 𝑥 add two.
And the other important thing we learnt in this video is the domain and range.
So we learnt the domain, they’re our input values and that’s always the 𝑥-axis.
And the range are our output values and that’s always the 𝑦-axis.
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