Year 11: Functions and relations

In Year 11 mathematics, functions and relations become a central language for describing how quantities are connected. Whether you are studying general mathematics, mathematical methods, or specialist mathematics, understanding these ideas now will make senior topics such as calculus, probability, and statistics much easier.

What is a relation?

At the most general level, a relation is simply a rule or connection between two sets. Usually we call the first set the domain and the second set the codomain. A relation tells us which elements from the domain are linked with which elements in the codomain.

A common way to think about a relation is as a set of ordered pairs: $(x, y)$ , where $x$ is from the domain and $y$ is from the codomain. If the pair $(x, y)$ is in the relation, we say that “x is related to y”.

Example: Favourite sport relation
Let the domain be a set of students: {Ava, Ben, Chloe}. Let the codomain be a set of sports: {Soccer, Netball}. A possible relation could be:

(Ava, Soccer)
(Ben, Soccer)
(Chloe, Netball)

This relation just records which sport is the favourite for each student.

Relations can be represented in many ways:

As a set of ordered pairs: ${(1, 2), (2, 3), (3, 5)}$
As a table of values
As a mapping diagram, with arrows drawn from domain elements to codomain elements
As a graph on the coordinate plane
Using an equation, such as $x ² + y ² = 9$

Domain, codomain and range

It is important to distinguish between three related ideas:

Domain: the set of all allowed input values (x-values).
Codomain: the set of possible outputs we are “targeting”.
Range (or image): the set of all actual output values that appear.

Example: Square relation
Consider the relation defined by the rule “y is the square of x”. If the domain is {−2, −1, 0, 1, 2}, the ordered pairs are:

(−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4)

If the codomain is “all real numbers”, then the range for this particular domain is {0, 1, 4}. Notice that the range is a subset of the codomain.

Key idea: The domain comes from how we choose or restrict inputs. The codomain is the set we consider as potential outputs. The range is what the relation actually produces.

When is a relation a function?

A function is a special type of relation. It is still a set of ordered pairs or a rule linking inputs and outputs, but with one important extra condition:

A relation is a function if and only if every input (x in the domain) is paired with exactly one output y.

In other words, in a function:

No input is left without an output.
No input has two different outputs.

Example: Is it a function?

Relation R₁ = {(1, 2), (2, 3), (3, 4)}.
Here each x-value (1, 2, 3) has exactly one y-value. R₁ is a function.
Relation R₂ = {(1, 2), (1, 3), (2, 5)}.
Here the input x = 1 is paired with two different outputs, 2 and 3. R₂ is not a function.

On a graph, we use the vertical line test to decide whether a relation is a function.

Vertical line test: A graph represents a function if and only if no vertical line cuts the graph at more than one point.

This works because a vertical line $x = a$ represents all points that share the same x-value. If we get more than one intersection, the input x = a would have more than one output y, which breaks the function rule.

Function notation

In Year 11, function notation becomes standard. Instead of writing “y = 2x + 3”, we often write: $f (x) = 2 x + 3$ .

In this notation:

$f$ is the name of the function.
$x$ is the independent variable, the input.
$f (x)$ means “the value of the function f at x”, which you can think of as the y-value.

Example: Evaluating function values
Let

f (x) = 3 x - 5

. Then:

$f (0) = 3 (0) - 5 = - 5$
$f (2) = 3 (2) - 5 = 1$
$f (- 1) = 3 (- 1) - 5 = - 8$

We can also write relations using function notation even if they are not really functions, but in school mathematics, if we write $f (x)$ we usually mean that it is a genuine function.

Different types of relations

It is useful to classify relations according to how multiple inputs and outputs connect. For a relation from set A to set B, we use words like:

One-to-one: every element of A is related to a unique element of B, and no two different elements of A share the same partner in B.
Many-to-one: two or more elements of A can be related to the same element in B.
One-to-many: one element of A is related to two or more elements in B.
Many-to-many: several elements of A may be related to several elements of B.

For a function from A to B:

It may be one-to-one or many-to-one.
It cannot be one-to-many or many-to-many, because that would give some x-values more than one y-value.

Example: Classifying functions

Let $f (x) = 2 x + 1$ on the domain of all real numbers. Different x-values produce different y-values, so this function is one-to-one.
Let $g (x) = x ²$ on the domain of all real numbers. Notice that $g (2) = 4$ and $g (- 2) = 4$ , so different inputs can give the same output. This is a many-to-one function.

Graphs of relations and functions

Graphs are a powerful way to visualise functions and relations. For a relation in the x–y plane, every allowed ordered pair (x, y) is plotted as a point. The shape formed by all these points shows how the variables relate.

For functions, we usually plot y = f(x). Some key ideas for Year 11 are:

Intercepts: where the graph crosses the axes.
End behaviour: what happens as x becomes very large or very negative.
Symmetry: whether the graph is symmetrical about an axis or a point.
Turning points: where the graph changes direction (important for quadratics and polynomials).

Example: Graph of a quadratic function
For

f (x) = x ² - 4

The graph is a parabola opening upwards.
It has x-intercepts where $x ² - 4 = 0$ , so $x = - 2$ or $x = 2$ .
It has a y-intercept at $x = 0$ , so $f (0) = - 4$ .
It is symmetric about the y-axis.

Restricting the domain and range

In many real-life problems, not every mathematical value makes sense. For example, time cannot be negative in most situations, and the number of people must be a whole number. In such cases, we restrict the domain or the range of a function.

Example: Height of a ball
A ball is thrown upwards and its height above the ground is modelled by

h (t) = - 5 t ² + 20 t + 1

, where

t

is time in seconds.

Mathematically, we could plug in any real number for t, but physically:

t ≥ 0, because time starts when the ball is thrown.
We usually stop the model when the ball hits the ground again, when h(t) = 0.

So the domain might be 0 ≤ t ≤ 4 (for example), and the range is the set of heights reached in that time interval.

Restricting the domain can also turn a relation that is not one-to-one into one that is one-to-one. This is important when you learn about inverse functions.

Inverse relations and inverse functions

The inverse of a relation is formed by swapping each ordered pair (x, y) to (y, x). Geometrically, this reflects the graph across the line y = x.

If the original relation is a function, the inverse relation might not be a function. However, if the original function is one-to-one (each y-value comes from exactly one x-value), then its inverse will also be a function. In Year 11, you may start to meet simple inverse functions.

Example: Inverse of a linear function
Let

f (x) = 3 x - 2

. To find its inverse:

Start with $y = 3 x - 2$ .
Swap x and y: $x = 3 y - 2$ .
Solve for y: $3 y = x + 2$ , so $y = (x + 2) / 3$ .

So the inverse function is

f ⁻ ¹ (x) = (x + 2) / 3

Relations, functions and real-world modelling

Relations and functions are not just abstract ideas; they are tools to describe real phenomena:

Physics: position as a function of time, velocity as a function of time.
Biology: population size as a function of time or resources.
Economics: cost as a function of quantity produced, profit as a function of price.
Everyday life: mobile data used as a function of time, temperature as a function of time of day.

When you build models, you will often need to:

Decide what the domain should be (what inputs make sense).
Check whether the relationship is reasonably described by a function.
Interpret key features of the graph (intercepts, turning points, asymptotes) in context.

Example: Taxi fare model
A simple taxi fare could be modelled as

F (d) = 4 + 2.5 d

, where F is the fare in dollars and d is the distance in kilometres. Here:

The domain might be d ≥ 0, since you cannot travel negative distance.
The graph is a straight line with y-intercept 4 and gradient 2.5.
The constant term 4 represents the flag fall (starting fee).
The coefficient 2.5 represents the cost per kilometre.

This is clearly a function: each distance gives exactly one fare.

Summary

In this tutorial, you have seen that:

A relation links elements from one set (domain) to another set (codomain).
A relation can be described using ordered pairs, tables, graphs, mapping diagrams or equations.
The domain is the set of allowed inputs, the range is the set of actual outputs, and the codomain is the set we think of as possible outputs.
A function is a relation in which each input has exactly one output.
We use function notation like $f (x)$ to describe functions more efficiently.
We classify relations as one-to-one, many-to-one, one-to-many, or many-to-many.
Only one-to-one and many-to-one relations can be functions.
Graphs help us visualise functions and relations, and we use the vertical line test to decide whether a graph represents a function.
We often restrict the domain and range to suit the real-world situation being modelled.
Inverse relations and inverse functions are formed by swapping inputs and outputs.

Mastering these ideas in Year 11 will give you a strong foundation for later topics such as exponential and logarithmic functions, trigonometric functions, and calculus in Year 12 and beyond.