Recipe

Find the domain, codomain, and range of a function

Use when: you're given a function in f: A \to B notation, with a rule, and asked for its domain, codomain, and range

Goal: the domain and codomain read off the notation, plus the range computed from the rule and checked against the codomain

Taught in Domain, codomain, range, image

1. 1

   In f: A \to B, the domain is A and the codomain is B, both announced before any output is computed.

   Example

   For f: \{1,2,3,4,5,6\} \to \mathbb{Z}: the domain is \{1,2,3,4,5,6\} and the codomain is \mathbb{Z}, both read straight from the notation.

   Why: domain and codomain are stated boundaries, not something to compute from the rule itself.

2. 2

   Apply the rule to each element of the domain one at a time, keeping track of every result.

   Example

   For f(n) = \lfloor n/2 \rfloor on \{1,2,3,4,5,6\}: f(1)=0, f(2)=1, f(3)=1, f(4)=2, f(5)=2, f(6)=3.

   • the domain is a small finite set — list f(a) for every single a, one at a time.
   • the domain is an infinite set described by a rule — describe the range with its own rule or interval, since an infinite list can't be finished.

   Why: the range only contains outputs that are actually produced, so it has to be found by running the rule, not read off the notation.

3. 3

   Write the range as the set of results from the previous step, dropping any repeat, since a set lists each value once.

   Example

   The raw outputs 0,1,1,2,2,3 collapse to the range \{0,1,2,3\}, with each repeated value counted only once.

   Why: repeated outputs from different inputs still count as a single element of the range.

4. 4

   Check that every value found in the previous step belongs to the codomain, since the range can equal the codomain or be a smaller subset of it.

   • the range equals the codomain — every announced target actually gets hit.
   • the range is a proper subset of the codomain — the codomain was announced larger than what the rule actually uses, which is still a legal codomain.

   Why: the codomain only promises an upper boundary, so checking this is what distinguishes it from the range instead of assuming the two always match.