A **function** is a relation between a set of inputs (called the **domain**), and a permissible set of outputs (called the **codomain**), such that each input is is related to exactly one output. We often denote a function on 1 variable by \( f\). When \( x\) is an element of the domain, we say that the output has the value \( f(x)\).

We can certainly add irrelevant items to the codomain, like {alligator}, {purple} and {Brilliant}. As such, we define the **range** of a function (also called the **image**) to be the set of all attained outputs. Note that by definition, the range of a function has to be a subset of the codomain.

Here are a few more terms that it is helpful to know:

**Injective**: An injective function is one that maps every value in the domain to a unique value in the codomain, such that for any given value in the range there is only one corresponding value in the domain. Injective functions are also called "one-to-one" functions.**Surjective**A surjective function is one that covers every element in the co-domain, such that there are no elements in the co-domain that are not a value of the function. In a surjective function the range and the codomain will be identical.**Bijective**A bijective function is both injective and surjective.

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