Classifying relations
A relation is described by its distinct inputs and outputs, then tested using the inputs. Repeated outputs are allowed in a function, but a repeated input with different outputs is not
- List distinct inputs and outputs The domain contains each input once and the range contains each output once \(\displaystyle R=\left\{(-2,3),(-2,5),(1,5)\right\}\Rightarrow D=\left\{-2,1\right\},\quad R_g=\left\{3,5\right\}\)
- Apply the input test Every input must be paired with exactly one output for the relation to be a function \(\displaystyle x=-2\Rightarrow y=3\text{ and }y=5\quad\Rightarrow\quad\text{not a function}\)
- Allow many-to-one functions Different inputs may share an output while each input still has one output \(\displaystyle \left\{(1,4),(2,4),(3,5)\right\}\Rightarrow\text{function and many-to-one}\)
- Use both repetition checks The correspondence records whether inputs and outputs are repeated \(\displaystyle \text{no repeated input or output}\Rightarrow\text{one-to-one},\quad\text{repeated input}\Rightarrow\text{one-to-many or many-to-many}\)
Worked example
The relation below is given by ordered pairs. State its domain and range, decide whether it is a function, and classify its correspondence
\[R=\left\{(-3,2),(-3,6),(1,6),(4,9)\right\}\]
- \(\displaystyle \operatorname{Dom}(R)=\left\{-3,1,4\right\}\)List each distinct input once
- \(\displaystyle \operatorname{Ran}(R)=\left\{2,6,9\right\}\)List each distinct output once, even though 6 appears twice
- \(\displaystyle x=-3\Rightarrow y=2\text{ and }y=6\quad\Rightarrow\quad R\text{ is not a function}\)The repeated input -3 has two different outputs
- \(\displaystyle -3\text{ is repeated and }6\text{ is repeated}\quad\Rightarrow\quad\text{many-to-many}\)Both inputs and outputs are repeated in the relation
Common mistakes
A function is required to be one-to-one
✕\(\displaystyle \left\{(1,4),(2,4),(3,5)\right\}\Rightarrow\text{not a function}\)✓\(\displaystyle \left\{(1,4),(2,4),(3,5)\right\}\Rightarrow\text{a many-to-one function}\)
A function allows different inputs to share one output as long as each input has one outputDomain and range are swapped
✕\(\displaystyle R=\left\{(-2,3),(-2,5),(1,5)\right\}\Rightarrow D=\left\{3,5\right\},\quad R_g=\left\{-2,1\right\}\)✓\(\displaystyle R=\left\{(-2,3),(-2,5),(1,5)\right\}\Rightarrow D=\left\{-2,1\right\},\quad R_g=\left\{3,5\right\}\)
The domain contains input coordinates and the range contains output coordinatesA repeated output is removed from the range
✕\(\displaystyle R=\left\{(1,4),(2,4),(3,5)\right\}\Rightarrow R_g=\left\{5\right\}\)✓\(\displaystyle R=\left\{(1,4),(2,4),(3,5)\right\}\Rightarrow R_g=\left\{4,5\right\}\)
A repeated output is still a range member, but it is listed only onceDifficulty levels
introClassify a small finite relation and decide whether it is a function
standardHandle repeated outputs and classify the full input-output correspondence
challengeRead restricted segments with open or closed endpoints and classify the relation
examSelect a compact classification statement containing the correct diagnostic reason
Questions about this topic
What is the difference between a function and a one-to-one function?+
A function gives each input exactly one output. A one-to-one function also has no repeated output from different inputs
How are repeated values written in a domain or range?+
Sets contain each member once, so a repeated input or output is written only once in the relevant set
How do open endpoints affect a relation's domain or range?+
An open endpoint is excluded from its segment, while a closed endpoint is included. The input or output value at an open endpoint can still belong to the domain or range if another point has that value
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 2: Functions and graphs
In this chapter
Read restricted graphs and horizontal level setsFind the gradient of a line through two pointsFind the equation of a line through two pointsSolve a pair of simultaneous linear equationsRead the vertical-axis intercept and gradient from an equationRead the point of intersection of two linesConstruct a linear model from a descriptionInterpret the graph of a linear modelTransform a quadratic and find its key featuresDetermine a quadratic rule from exact informationAnalyse signs and solve a quadratic inequalityAnalyse a cubic graph or determine its ruleInterpret and determine a transformed power functionConnect a factorised higher-degree rule and its graphSolve a higher-degree polynomial equation or inequalityApproximate solutions from intersections of two graphsClassify a relation and state its domain and rangeEvaluate and determine function rulesApply and describe transformations of a functionAnalyse the general rational-power familyFind the maximal domain of an algebraic functionAnalyse reciprocal and square-reciprocal functionsAnalyse sideways-parabola and root-function familiesDetermine a power-family rule from features and pointsWork with sets and interval notationSimplify, compare and operate with surdsCombine function rules and ordinatesRecognise equivalent polynomial formsUse a piecewise linear modelEvaluate exact circular-function values by symmetry and periodicityKey points and features of the basic sine and cosine graphsTransform and reconstruct sine and cosine functionsFeatures and transformations of tangent functionsForm and use periodic sine and cosine modelsAnalyse exponential and logarithmic graphsInterpret and use an exponential modelSelect and interpret a function modelEvaluate and interpret a restricted rational model