Domains, Codomains, and Ranges

That is, the domain of a function \(f\) is the set of all objects/elements that have a corresponding output through \(f\). In other words, it’s the set of all elements that you can “plug into” \(f\).

In other words, the codomain of a function, in a sense, represents the type of output we can expect to get from \(f\); not specifically what the outputs of the function are. The set of specific outputs of a function is what define next.

An Example of Domain, Codomain, and Range Using a Potato Diagram

Let \(f\) be defined by the potato diagram below.

The set of all inputs that have a corresponding output via \(f\) is \(\text{dom}(f)=\{1,2,3,5\}\).

The range of \(f\) is \(\text{ran}(f)=\{-1,0,3,4\}\); i.e. the set of all possible outputs of \(f\) (in this case, the set of all numbers with arrows pointing to them).

The codomain of \(f\) is \(\text{codom}(f)=\{-1,0,3,4,5\}\), which is the set to which all outputs of \(f\) are constrained. No output of \(f\) will fall outside the codomain given just now.

Example with a Function Defined By a Rule

Let \(f\) be the function defined as \(f(x)=x^2\). Note that \(\text{dom}(f)=\mathbb{R}\) (or at least can be assumed so), because there is no real number that we cannot plug into \(f\); i.e. we can plug in whatever real number we like and we will get out an answer. We know that \(f\) will only produce real number outputs, so the codomain of \(f\) is \(\text{codom}(f)=\mathbb{R}\). Finally, the range of \(f\) is \(\text{ran}(f)=[0,\infty)\), because the outputs (i.e. the \(y\)-values of \(f\)) are all greater than or equal to \(0\) (you can’t get a negative number by squaring something).

How Functions are Sometimes Defined

Occasionally, when defining a function, we outright list the domain and codomain of the function so that it is clear what sets the function is mapping from and to. Suppose for example that we are defining a function from the set \(A\) to the set \(B\). We would then write “Let \(f:A\rightarrow B\) be a function defined such that…” to define the function, where the first set \(A\) is the domain and the second set \(B\) is the codomain. We could easily just skip on the notation and define \(f\) by saying “Let \(f\) be a function from \(A\) to \(B\) be defined…” and then list how the function is defined (i.e. by a rule, diagram, graph, etc.).

Point of Interest: You might be wondering why we even bother with codomains of functions when we could write the range set instead. One reason for this is that, for complicated functions, determining what the range is can be a horribly cumbersome task. Its often easier to state what type of output we could expect.