Grades 6–8FoundationsLesson available

Functions

A function assigns to each input in its domain exactly one output in its codomain. Arrow diagrams, the set-of-ordered-pairs definition, images, and the properties injective, surjective and bijective.

math.found.functions

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Builds on

Sets Integers

Required prerequisites, all the way down: 6

You will learn to

Decide whether a rule, table or arrow diagram defines a function from a given domain to a given codomain.
Identify the domain, codomain and image of a function, and explain why the image can be smaller than the codomain.
Describe a function as a set of ordered pairs and classify functions as injective, surjective or bijective.
Count the functions from a finite set A to a finite set B, and the injections when |A| ≤ |B|.

Definition

A function f : A → B assigns to each element a of the domain A exactly one element f(a) of the codomain B.
Formally, a function from A to B is a subset f ⊆ A × B such that for each a ∈ A there is exactly one b ∈ B with (a, b) ∈ f; the domain and codomain are part of the data.
The image of f is the set {f(a) : a ∈ A}, a subset of the codomain.
f is injective if f(a) = f(a′) implies a = a′, surjective if every b ∈ B equals f(a) for some a ∈ A, and bijective if it is both.

Theorem

There are |B|^|A| functions from a finite set A to a finite set B; in particular there is exactly one function from ∅ to any set.
If |A| = m ≤ n = |B|, there are n(n − 1)⋯(n − m + 1) injections from A to B.
For finite sets with |A| = |B|, a function A → B is injective if and only if it is surjective.
A function has an inverse function exactly when it is bijective.

Common misconceptions

Thinking a function must be given by a formula; any rule assigning exactly one output to each input is a function.
Believing every element of the codomain must be an output; that is surjectivity, not part of the definition.
Thinking a function may not send two different inputs to the same output.
Ignoring the codomain: n ↦ 2n is surjective as a function from ℤ to the even integers, but not as a function from ℤ to ℤ.

Leads to

Linear relationships and simultaneous equations Function machines in sequence Countability and Cantor's diagonal argument Trigonometry Functions and their graphs (precalculus)Composition Algebraic structures: monoids, groups and rings

Optional discoveries

Products by maps: a first universal property A function into a Cartesian product A × B is exactly a pair of functions, one into A and one into B: the first example of a universal property.Wonder Trail →
Programming for mathematics Programming languages also have 'functions', but a program function may read input, change state or fail to return, so it need not be a mathematical function.