A function is one relationship seen in several forms. A table, graph, equation, mapping diagram and word description may look different, but they can all encode the same rule connecting an input to an output.
This Secondary 2 Mathematics Learning Guide develops functions as a unifying idea rather than another isolated chapter. The learner will identify inputs and outputs, read domain restrictions, move among equations, tables and graphs, distinguish a function rule from a list of values, solve forward and reverse questions, and verify that several representations really describe the same relationship.
Secondary Mathematics Hub: S1–S4 Capability Map · Secondary 2 Learning Guide, Batch 7, Guide 1. Companion guides cover mathematical modelling and model limits, dimensional reasoning and conversion chains, and problem posing and reverse engineering.
Course boundary. Function language develops through lower and upper secondary. This guide uses familiar linear and simple quadratic examples to connect representations. It does not replace the dedicated guides on linear graphs or quadratic functions; its job is to show what remains invariant when the representation changes.
Navigate: what is a function? · domain · tables · graphs · equations · reverse questions · multiple representations · practice and answers · teaching and transfer.
1. A function assigns one output to each permitted input
Suppose the rule is y = 3x + 2. If x = 4, the output is y = 14. If x = −1, the output is y = −1. Each permitted x-value produces exactly one y-value.
This is the central function idea: for every input in the domain, the rule produces one output. Different inputs are allowed to share the same output. What is not allowed in an ordinary function is one single input producing two different outputs under the same rule.
Worked example 1: function machine
A machine multiplies an input by 5, then subtracts 3. If x is the input, output y = 5x − 3. Inputs 0, 1 and 4 give outputs −3, 2 and 17.
The verbal instruction, algebraic equation and list of input-output pairs are three representations of one function.
Worked example 2: a non-function relation
Suppose a relation lists (2,5) and (2,8). The same input x = 2 is paired with two outputs. Unless an additional variable or condition distinguishes the cases, this relation is not a function of x.
The problem is not that two points exist. The problem is that one input has conflicting outputs.
2. The domain tells us which inputs are permitted
The domain is the set of inputs for which the rule is being used. Sometimes the domain is stated explicitly. Sometimes it is restricted by the formula or the context.
For y = 10/(x − 2), x = 2 is excluded because the denominator would be zero. For a model where n counts students, the domain may be non-negative whole numbers rather than all real numbers.
Worked example 3: formula restriction
For f(x) = 6/(x + 1), the input x = −1 is not permitted. At x = 2, f(2) = 6/3 = 2. The rule has meaning only on its permitted domain.
Worked example 4: contextual domain
A hypothetical ticket model is C = 12n, where n is the number of tickets. Algebraically, n = 2.5 could be substituted. Contextually, ordinary indivisible tickets require whole-number n. The function rule is being used on a discrete domain.
Domain is therefore part of mathematical meaning, not an afterthought added after graphing.
3. A table samples the function at selected inputs
A table does not usually show every possible input. It shows selected cases. For y = 2x + 1:
| x | y |
|---|---|
| −2 | −3 |
| −1 | −1 |
| 0 | 1 |
| 1 | 3 |
| 2 | 5 |
The constant change in y is 2 when x increases by 1. This reveals the linear rate of change.
Worked example 5: infer a linear rule from a table
A table gives x = 0, 1, 2, 3 and y = 4, 7, 10, 13. The output increases by 3 per unit input, so the gradient is 3. When x = 0, y = 4, so the rule is y = 3x + 4.
Check x = 3: 3(3) + 4 = 13. The rule reproduces the table.
A table can hide behaviour between sampled values
If only three table entries are supplied, several different rules might pass through them. The intended function family, context or graph may provide additional constraints. A finite table is evidence about a rule; it may not uniquely determine every possible rule without more information.
4. A graph shows how outputs behave across the domain
Each point (x,y) on a function graph represents an input-output pair satisfying the rule. A straight line shows constant rate of change. A parabola shows a changing rate and may have a turning point. A graph therefore reveals behaviour that a short table may hide.
Worked example 6: read from a line
For y = 2x + 5, the graph crosses the y-axis at 5 and rises by 2 for each 1 unit increase in x. The point (3,11) lies on the graph because 11 = 2(3) + 5.
The equation verifies a plotted point; the graph visualises the equation’s whole family of solutions.
Worked example 7: vertical-line reasoning
If a vertical line x = 4 cuts a plotted relation at two points, that relation assigns two y-values to input 4 and therefore is not a function of x. This is the idea behind the vertical-line test.
A vertical line itself, x = 4, is not a function y = f(x), because input x = 4 corresponds to many y-values.
5. An equation compresses the relationship into a rule
The equation y = mx + c tells us two key linear features immediately: m is the rate of change and c is the output when x = 0.
For y = −3x + 8, every increase of 1 in x decreases y by 3, and the y-intercept is 8. These same facts can be read from a table or graph.
Worked example 8: compare two equations
Function A: y = 4x. Function B: y = 4x + 7. Both have rate of change 4. Only A is directly proportional because y/x is constant and its graph passes through the origin.
Same gradient does not mean same function. The fixed term changes every output.
Worked example 9: simple quadratic function
For y = x² − 4, inputs −3, −2, −1, 0, 1, 2, 3 give outputs 5, 0, −3, −4, −3, 0, 5. The function gives the same output for some different inputs, such as x = −3 and x = 3.
This is still a function because each individual input has only one output.
6. Reverse questions ask which input produced a given output
If y = 5x + 2 and y = 37, solve 37 = 5x + 2. Then 35 = 5x and x = 7. We have reversed the function calculation by solving an equation.
For some functions, one output may correspond to more than one input. For y = x², output 9 comes from x = 3 and x = −3. The function is valid forward, but its reverse relation is not single-valued unless the domain is restricted.
Worked example 10: reverse a linear function
Given f(x) = 3x − 4, find x when f(x) = 20. Solve 3x − 4 = 20, giving x = 8.
Worked example 11: reverse with domain information
Suppose g(x) = x² for x ≥ 0. If g(x) = 25, then x = 5. The domain excludes −5, making the reverse answer unique.
7. Multiple representations should agree on the same test cases
If a word description says “start at 10 and add 4 per unit,” the equation should be y = 4x + 10 if x = 0 corresponds to the starting point. A table should contain (0,10), (1,14), (2,18). A graph should pass through those points with gradient 4.
When one representation disagrees, do not choose the one that looks most familiar. Test a common input and trace where the disagreement begins.
Worked example 12: detect a mismatched graph description
A table gives y-values 7, 10, 13 for x = 0, 1, 2. A proposed equation is y = 3x + 4. At x = 0 this gives 4, not 7. The proposed equation cannot represent the table. The correct rule is y = 3x + 7.
Testing x = 0 is efficient because it isolates the intercept immediately.
8. Functions connect several Secondary 2 topics
- Ratio and direct proportion: y = kx.
- Rate: gradient measures change per unit input.
- Linear graphs: tables, equations and coordinates describe the same line.
- Quadratic functions: one rule can produce symmetric outputs and a turning point.
- Sequences: position n can act as function input.
- Formulae: substituting values evaluates a function-like relationship.
- Modelling: a function can encode an assumption about how one quantity depends on another.
9. Common function errors
- Input and output swapped: repair variable ownership.
- One input paired with two outputs unnoticed: repair function definition.
- Domain restriction ignored: repair permitted-input reasoning.
- Table difference mistaken for intercept: repair representation mapping.
- Same gradient treated as same line: repair intercept awareness.
- Reverse quadratic question given one root without domain reason: repair solution-set awareness.
- Graph read by eye instead of coordinates/scale: repair representation precision.
- Finite table treated as unique infinite rule without assumptions: repair model-family awareness.
10. Practice: move among representations
Questions 1–6. 1. For y = 4x − 1, find y when x = 3. 2. Find y when x = −2. 3. Which input is excluded from y = 5/(x − 7)? 4. Is the relation {(1,2),(2,4),(1,5)} a function of x? Explain. 5. For y = 3x + 6, state the output when x = 0. 6. State the rate of change.
Questions 7–12. 7. A table has x = 0,1,2,3 and y = 5,9,13,17. Find a linear rule. 8. Use it to find y when x = 20. 9. Find x when y = 61. 10. For y = x² − 1, find outputs for x = −2,0,2. 11. Is the rule a function? 12. Why can output 3 correspond to two inputs?
Questions 13–18. 13. Compare y = 2x and y = 2x + 5: what is the same and what differs? 14. Which is directly proportional? 15. A model says “fixed charge 8 plus 3 per unit.” Write an equation. 16. Produce the first three outputs for x = 0,1,2. 17. A proposed table gives 8,12,14. Which entry breaks the model? 18. State one way to verify that an equation and graph represent the same function.
Explained answers 1–6
1. 11. 2. −9. 3. x = 7. 4. No; input 1 has outputs 2 and 5. 5. 6. 6. 3 units of output per unit input.
Explained answers 7–12
7. Difference 4 and y-intercept 5 give y = 4x + 5. 8. 85. 9. 4x + 5 = 61 gives x = 14. 10. 3, −1, 3. 11. Yes; each input has one output. 12. Different inputs may share an output; here x = ±2 both give 3.
Explained answers 13–18
13. Both have gradient 2; intercepts are 0 and 5. 14. y = 2x. 15. y = 3x + 8. 16. 8, 11, 14. 17. The second output 12 should be 11. 18. Test several x-values and confirm the plotted points satisfy the equation, including x = 0 and another non-zero value.
11. Teaching sequence: one relationship, five representations
Choose one simple rule such as y = 3x + 2. Ask the learner to describe it in words, build a table, plot points, identify gradient and intercept, then solve a reverse question. The aim is to make each representation translate into the others without losing the relationship.
Next change only one feature: add a domain restriction, replace the line with a simple quadratic, or use a discrete context. Ask which representations change and which ideas remain the same.
Questions parents and tutors can ask
What is the input? What is the output? Which inputs are allowed? How does the output change when the input increases by one? Where do you see that in the table, equation and graph? Can two inputs share one output? Can one input have two outputs here?
12. The transfer test: preserve the function while changing the story
A fictional storage model is V = 5n + 20. A transport model is C = 5d + 20. A sequence model is Tn = 5n + 20. The letters and contexts differ, but all three share the same linear function structure.
At input 10, each output is 70. Each has rate of change 5 and intercept 20. A learner who recognises this common structure can transfer methods without waiting for the chapter label.
Identify the input and output. Respect the domain. Translate among words, tables, equations and graphs. Solve forward and reverse questions. Verify that every representation preserves the same relationship.
Continue to Mathematical Modelling, Assumptions, Validation and Model Limits · Return to the Secondary Mathematics Hub.
