Functions — Definition, Domain, Codomain and Range

A function is a special kind of relation, obeying one extra rule that makes it far more useful for describing dependable, predictable relationships — such as how a firm's total cost depends on the quantity it produces.

A relation $f$ from a set $A$ to a set $B$ is called a function if every element of $A$ is related to exactly one element of $B$ — never zero elements, and never more than one. We write $f: A \to B$, and if $a \in A$ is paired with $b \in B$ under $f$, we write $b = f(a)$ and call $b$ the image of $a$ under $f$.

Three sets go with every function $f : A \to B$:

• The domain is the set $A$ itself — every element of $A$ must have an image, with no exceptions.
• The codomain is the set $B$ — the set the images are drawn from, whether or not every element of $B$ is actually used.
• The range (or image set) is the subset of $B$ consisting only of those elements that are actually the image of some element of $A$. The range is always a subset of the codomain, and the two are equal only in a special case (taken up as 'onto' functions in the next section).

A convenient way to test whether a given relation, described by an arrow diagram or a graph, is a function is the vertical line test: if a graph is drawn with $x$ along the horizontal axis and $y=f(x)$ along the vertical axis, then the relation is a function of $x$ if and only if every vertical line meets the graph in at most one point. A circle, for instance, fails this test (a vertical line through the middle of the circle meets it twice) and is therefore not the graph of a function of $x$. …