Determine the range of 𝑓 of 𝑥.
We have a diagram in this question. This is the mapping diagram of the function 𝑓 of 𝑥. The function takes inputs from the set 𝑥 and returns outputs from the set 𝑦. 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.
In the question, we have to find the range of 𝑓 of 𝑥. So let’s review the meaning of the range and related terms. The set of inputs to a function is called the domain of that function. We can see that the set of inputs to our function are negative eight, 14, and negative two. So in our case, this is the set containing negative eight, 14, and negative two, which we can recognise as the set 𝑥.
The set of outputs of a function is called the range of that function. This is what we’re asked to find in the question. You might think that the set that we’re looking for is the set 𝑦. But, take a look at the element 18 of the set 𝑦. It’s in the set. But, it isn’t an output of the function. There is no input for which 𝑓 of 𝑥 is 18. The function has only two possible outputs, negative 20 and negative four. These are the two elements of the set 𝑦 which have arrows pointing at them.
The range is a set containing these two outputs, the set of negative 20 and negative four. This is a subset of the set 𝑦 but not the entire set. The set 𝑦 in this context does in fact have a name. It’s called the codomain of the function. And it’s true in general that the range is always a subset of the codomain. But, as we’ve seen, the range doesn’t have to be equal to the codomain.
So to conclude, the range of the function, which was what the question was asking for, is the set negative 20, negative four. These are the values that have arrows pointing at them in our mapping diagram. The domain of the function is the set negative eight, 14, and negative two. Those can be written in any order of course. And they are the inputs to the function which have arrows coming out of them. And the set 𝑦 which contains 18, 15, negative 20, negative four, and 16 is not the range of the function. It’s the codomain of the function.