Functions — Pictorial Representation, Domain, Co-domain and Range

A function from a set $A$ to a set $B$ is a rule that assigns to every element of $A$ exactly one element of $B$. This syllabus studies functions entirely through their pictorial representation — an arrow diagram — rather than through algebraic formulas, so the two conditions above translate directly into how the diagram must look:

• Every element of $A$ must have an arrow leaving it (no element of $A$ is left unmapped).
• Exactly one arrow leaves each element of $A$ (no element of $A$ has two or more arrows going to different elements of $B$).

If either condition fails — some element of $A$ has no arrow, or some element of $A$ has more than one arrow — the diagram represents a relation, but not a function.

Worked Example 10, part (i) — a Function:

Every element of $A$ has exactly one outgoing arrow (note $q$ receives arrows from both $2$ and $4$, which is allowed) — this is a function.

Worked Example 10, part (ii) — Not a Function:

Element $2$ has two outgoing arrows and element $4$ has none — either fault alone is enough to disqualify it, so this diagram is not a function.

A function's domain is the starting set $A$ — every element of the domain must be used exactly once. Its co-domain is the set $B$ it maps into — this is simply the set the arrows are allowed to land in, whether or not every element of $B$ actually gets an arrow. Its range is the set of elements of $B$ that actually receive at least one arrow — the range is always a subset of the co-domain, but need not equal it. …