Relations, Domain And Range
Understand relations, domain and range for Queensland Year 11 Mathematical Methods (QCAA). A relation maps one set to another as ordered pairs, linking an independent input variable to a dependent output variable.
You will learn to state the domain and range from ordered pairs, from a graph with open and closed endpoints, and as the maximal domain of a rule — essential groundwork before you meet functions.
Every question with a fully worked solution.
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Theory
In Year 11 Mathematical Methods (QCAA, Unit 1) a relation is any set of ordered pairs \((x,y)\) — a rule linking two variables. Its domain is the set of allowed \(x\)-values and its range is the set of resulting \(y\)-values. This page shows how to state the domain and range from a list of points, read them off a graph with open and closed endpoints, and find the maximal domain from a rule.
A relation is a set of ordered pairs \((x,y)\); it can be given as a list of points, a graph, or a rule. The domain is the set of all first coordinates (the \(x\)-values), and the range is the set of all second coordinates (the \(y\)-values).
When points share the same \(x\) or \(y\)-value, that value is listed only once in the set. On a graph, a closed dot (filled) means the endpoint is included, and an open dot (hollow) means it is excluded — exactly matching square and round brackets in interval notation.
The maximal (implied) domain is the largest set of \(x\)-values for which the rule gives a real \(y\). Watch for a square root (the inside must be \(\ge0\)) and a denominator (which cannot be \(0\)).
For a finite relation \(R=\{(x_1,y_1),\dots,(x_n,y_n)\}\):
Maximal-domain conditions for the common rules:
How to state domain and range
- Identify the form — a list of points, a graph, or a rule.
- Domain: collect every \(x\)-value (points), sweep left-to-right (graph), or apply the maximal-domain conditions (rule), listing each value once.
- Range: collect every \(y\)-value or sweep bottom-to-top, then write both in interval or set notation, matching open/closed ends to round/square brackets.
Domain — list every first coordinate, each once:
| \(\text{dom}\) | \(=\) | \(\{-2,\,0,\,1,\,1\}\) |
| \(=\) | \(\{-2,\,0,\,1\}\) |
The input \(1\) appears twice but is written once.
Range — list every second coordinate:
| \(\text{ran}\) | \(=\) | \(\{3,\,1,\,4,\,-2\}\) |
| \(=\) | \(\{-2,\,1,\,3,\,4\}\) |
Domain \(\{-2,0,1\}\); range \(\{-2,1,3,4\}\).
Domain — sweep left-to-right along the \(x\)-axis:
| \(x\) | \(\in\) | \([-3,\,2)\) |
Closed at \(-3\) (square bracket), open at \(2\) (round bracket).
Range — the \(y\)-values at the two ends:
| \(y(-3)\) | \(=\) | \(-3+1=-2\) |
| \(y(2)\) | \(=\) | \(2+1=3\) |
| \(\text{ran}\) | \(=\) | \([-2,\,3)\) |
Domain \([-3,2)\); range \([-2,3)\).
Domain — the expression under the root must be \(\ge0\):
| \(x-2\) | \(\ge\) | \(0\) |
| \(x\) | \(\ge\) | \(2\) |
So the maximal domain is:
| \(\text{dom}\) | \(=\) | \([2,\,\infty)\) |
Range — a square root is never negative and grows without bound:
| \(\sqrt{x-2}\) | \(\ge\) | \(0\) |
| \(\text{ran}\) | \(=\) | \([0,\,\infty)\) |
Domain \([2,\infty)\); range \([0,\infty)\).
Check the turning point — the vertex of \(y=x^2-4\) is at \(x=0\), inside the domain:
| \(y(0)\) | \(=\) | \((0)^2-4\) |
| \(=\) | \(-4\) |
This is the lowest value (the minimum).
Test the endpoints for the highest value:
| \(y(-1)\) | \(=\) | \((-1)^2-4=-3\) |
| \(y(3)\) | \(=\) | \((3)^2-4=5\) |
Range runs from the minimum up to the largest endpoint value:
| \(\text{ran}\) | \(=\) | \([-4,\,5]\) |
Range \([-4,5]\).
Common pitfalls
Frequently asked questions
What is the difference between domain and range?
The domain is the set of all \(x\)-values (inputs) of a relation; the range is the set of all \(y\)-values (outputs). Read the domain left-to-right and the range bottom-to-top.
How do you find the domain and range from a set of points?
List every first coordinate for the domain and every second coordinate for the range, writing each repeated value only once.
What do open and closed dots mean on a graph?
A closed (filled) dot means the endpoint is included — a square bracket. An open (hollow) dot means it is excluded — a round bracket.
How do you find the maximal domain of a function?
Find every \(x\) that gives a real value: the inside of a square root must be \(\ge0\) and a denominator cannot be \(0\). All other \(x\) are allowed.
Why is the minimum of the range sometimes not at an endpoint?
If the graph has a turning point inside the domain, the smallest (or largest) \(y\) occurs there. For \(y=x^2-4\) on \([-1,3]\) the minimum is at the vertex \(x=0\).
How do you write domain and range in interval notation?
Use a square bracket for an included end and a round bracket for an excluded end, and \(\infty\) always with a round bracket — for example \([2,\infty)\).