A function is one of the most important organising ideas in Mathematics because it connects numbers, algebra, graphs, tables, formulas and real-world relationships. Yet many students first meet functions as notation and never build the underlying picture.
This guide develops functions as a rule for assigning exactly one output to each permitted input. It explains what that sentence really means, why one input cannot produce two different outputs in the same function, how functions appear in tables and graphs, and why function thinking becomes a bridge into later algebra and Additional Mathematics.
One-sentence answer
A function is a relationship that assigns each allowed input exactly one output.
Input, rule, output
Consider the rule “multiply by 2 and add 3”. If the input is 4, the output is 11. If the input is -1, the output is 1.
x → 2x + 3The same rule acts on every permitted input. The function is not the answer 11. The function is the relationship that turns an input into an output.
Function notation
Instead of repeatedly writing “the output when the input is x”, Mathematics writes:
f(x) = 2x + 3Then:
f(4) = 2(4) + 3 = 11The symbol f is the name of the function. The expression f(4) means “the output of function f when the input is 4”.
Why one input must have one output
Suppose a rule says that input 3 sometimes produces 7 and sometimes produces 12, with no additional information deciding which. That is not a well-defined function of that input alone.
Different inputs may produce the same output. That is allowed. For example, f(x)=x² gives f(2)=4 and f(-2)=4. The restriction is the other direction: one input cannot be assigned two different outputs by the same function.
Functions in tables
| x | f(x)=2x+3 |
|---|---|
| -1 | 1 |
| 0 | 3 |
| 1 | 5 |
| 2 | 7 |
| 3 | 9 |
The table shows selected input-output pairs. The formula describes the relationship for every permitted input, not just the values listed.
Functions in graphs
A graph of y=f(x) displays input-output pairs as points. The horizontal coordinate is the input x; the vertical coordinate is the output y.
A useful graphical test is the vertical line test. If a vertical line crosses a graph more than once, then one x-value is paired with more than one y-value, so the graph does not represent y as a function of x.
Worked example 1: evaluate a function
Let f(x)=3x-5. Find f(6).
f(6)=3(6)-5=18-5=13The answer is 13.
Worked example 2: find an input from an output
Let f(x)=4x+1. Find x when f(x)=21.
4x+1=21
4x=20
x=5This reverses the input-output question using algebra.
Worked example 3: same output, different inputs
Let g(x)=x². Find two inputs that give output 9.
x²=9
x=3 or x=-3This does not break the function rule. Each input still has exactly one output. It shows that a function can be many-to-one.
Functions as models
If a taxi fare is modelled by C(d)=4+0.80d, where d is distance travelled, the function encodes a relationship: each permitted distance produces a cost according to the model.
The formula is useful because it can generate many outputs, be graphed, compared with another pricing rule, or solved backwards for distance from cost.
Functions are not only formulas
A function can be described by a formula, table, graph, mapping diagram, algorithm or verbal rule. These are different representations of the same underlying assignment idea.
Strong function thinking means moving between representations without losing the relationship.
Diagnostic table
| Observed mistake | Likely issue | Repair |
|---|---|---|
| Treats f(x) as f × x | Notation meaning | Read f(x) aloud as “f of x” |
| Thinks two inputs cannot share an output | Direction of function rule | Use x² with inputs ±2 |
| Only works from formulas | Representation dependency | Translate among table, graph and rule |
| Substitutes incorrectly into brackets | Algebra execution | Write the substituted expression before simplifying |
A tutor’s teaching sequence
- Begin with an input-output rule in words.
- Build a short table.
- Introduce f(x) notation only after the relationship is clear.
- Plot the corresponding points.
- Reverse a few outputs to recover inputs.
- Change representation and test whether the learner preserves the same function.
Practice
- For f(x)=5x-2, find f(3).
- For g(x)=x²+1, find g(-4).
- If h(x)=3x+7 and h(x)=25, find x.
- Can two different inputs in a function produce the same output?
- Why does a graph failing the vertical line test not represent y as a function of x?
Answers
1. 13. 2. 17. 3. 6. 4. Yes. 5. Because at least one input x would have more than one output y.
Connected routes
Continue to Input, Output and Function Machines as Early Function Thinking, Domain, Range and Restrictions, and Composite and Inverse Thinking Before Additional Mathematics. Return to the Mathematics Learning Hub for the wider Secondary Mathematics route.