WEBVTT
00:00:00.360 --> 00:00:03.360
Determine the range of 𝑓 of 𝑥.
00:00:04.400 --> 00:00:06.920
We have a diagram in this question.
00:00:07.400 --> 00:00:10.560
This is the mapping diagram of the function 𝑓 of 𝑥.
00:00:11.320 --> 00:00:17.720
The function takes inputs from the set 𝑥 and returns outputs from the set 𝑦.
00:00:18.440 --> 00:00:29.520
For example, by looking at the highlighted arrow of the mapping diagram, we can see that the value of 𝑓 of negative eight is negative 20.
00:00:30.640 --> 00:00:36.200
In the question, we have to find the range of 𝑓 of 𝑥.
00:00:36.200 --> 00:00:39.840
So let’s review the meaning of the range and related terms.
00:00:40.480 --> 00:00:44.360
The set of inputs to a function is called the domain of that function.
00:00:45.000 --> 00:00:54.240
We can see that the set of inputs to our function are negative eight, 14, and negative two.
00:00:55.160 --> 00:01:07.680
So in our case, this is the set containing negative eight, 14, and negative two, which we can recognise as the set 𝑥.
00:01:08.200 --> 00:01:11.920
The set of outputs of a function is called the range of that function.
00:01:12.440 --> 00:01:15.080
This is what we’re asked to find in the question.
00:01:16.000 --> 00:01:20.720
You might think that the set that we’re looking for is the set 𝑦.
00:01:21.280 --> 00:01:27.200
But, take a look at the element 18 of the set 𝑦.
00:01:27.800 --> 00:01:30.000
It’s in the set.
00:01:30.600 --> 00:01:33.080
But, it isn’t an output of the function.
00:01:33.760 --> 00:01:41.120
There is no input for which 𝑓 of 𝑥 is 18.
00:01:42.000 --> 00:01:48.320
The function has only two possible outputs, negative 20 and negative four.
00:01:48.920 --> 00:01:53.440
These are the two elements of the set 𝑦 which have arrows pointing at them.
00:01:53.920 --> 00:02:03.360
The range is a set containing these two outputs, the set of negative 20 and negative four.
00:02:04.160 --> 00:02:10.160
This is a subset of the set 𝑦 but not the entire set.
00:02:10.680 --> 00:02:15.200
The set 𝑦 in this context does in fact have a name.
00:02:15.880 --> 00:02:19.120
It’s called the codomain of the function.
00:02:19.520 --> 00:02:25.320
And it’s true in general that the range is always a subset of the codomain.
00:02:25.960 --> 00:02:31.800
But, as we’ve seen, the range doesn’t have to be equal to the codomain.
00:02:32.640 --> 00:02:42.360
So to conclude, the range of the function, which was what the question was asking for, is the set negative 20, negative four.
00:02:42.560 --> 00:02:48.080
These are the values that have arrows pointing at them in our mapping diagram.
00:02:48.600 --> 00:02:56.280
The domain of the function is the set negative eight, 14, and negative two.
00:02:56.680 --> 00:02:59.720
Those can be written in any order of course.
00:03:00.200 --> 00:03:03.880
And they are the inputs to the function which have arrows coming out of them.
00:03:04.560 --> 00:03:16.800
And the set 𝑦 which contains 18, 15, negative 20, negative four, and 16 is not the range of the function.
00:03:17.320 --> 00:03:19.680
It’s the codomain of the function.