Function Basics

Domain, range, and injective / surjective / bijective — the language of functions.

The idea

Definition (Function).

A function $f \colon A \to B$ is a rule that assigns to each element of a set $A$ exactly one element of a set $B$: no input goes without an output, and none receives two.

A number's square, a time's temperature, a shape's area are all functions.

The set $A$ of allowed inputs is the domain. The set $B$ is the target, and the outputs the rule actually produces, $\{f(x) : x \in A\}$, form the range. The range can be smaller than the target: a function may aim at all of $\mathbb{R}$ and land on only part of it. When a formula arrives with no domain stated, we take its natural domain, every real input the formula makes sense for; division by zero and even roots of negatives are the usual exclusions.

Three words describe how a function matches its two sets. It is injective, or one-to-one, when distinct inputs give distinct outputs; surjective, or onto, when its range is the whole target; and bijective when it is both, so that every element of $B$ comes from exactly one element of $A$.

Ways to work on it

• Walkthrough. Domain, range, and the one-to-one / onto / bijective vocabulary.
• Practice. Classify a random function: domain, range, injective / surjective / bijective.
• Hardest. Subtler domain edge cases and injective / surjective / bijective reasoning.

Not sure where to start? Take the ten-question placement test.