If 𝑓 is a function from the set 𝑋 to the set 𝑌, what do we call 𝑌?
Let’s begin by reminding ourselves what it means for 𝑓 to be a function from one set to another. If 𝑓 is a relation, it takes values from set 𝑋 and maps them into values in set 𝑌. If, however, 𝑓 is a function, any value in set 𝑋 maps onto exactly one value in set 𝑌. So, let’s try and picture what could happen here. Suppose set 𝑋 contains the elements one, two, three, and five and set 𝑌 contains the elements seven, nine, 11, 13, and 19. If 𝑓 is a function, any element in set 𝑋 maps onto exactly one in set 𝑌. So, element one in 𝑋 maps onto seven in 𝑌. Element two in 𝑋 could map onto seven in 𝑌. Element three maps onto 13. And finally, element five maps onto 19.
Then the set of possible values that we can input to the function is said to be the domain. Then the set of possible values that are output from the function 𝑓 is said to be the range. Here that’s seven, 13, and 19. But if we look carefully, we notice that set 𝑌 contains two extra values. It contains nine and 11. So, set 𝑌 can’t actually be the range. The range is the set of values that actually come out of the function, whilst set 𝑌 contains other possible values. In this case, set 𝑌 is said to be the codomain of the function 𝑓. And so there’s a slight difference between the range and codomain.
Given the domain of the function, the codomain is the set of values that could come out, whilst the range is the set of values that does come out of the function. And so that allows us to answer the question. What do we call set 𝑌? We call it the codomain of the function 𝑓.