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Module 1 · L1 of 15 ~35 min ⚡ +50 XP in Learn · +25 to complete

Functions & Relations

Every time your phone recognises your face, it relies on a simple mathematical rule: one input, one output. That's the essence of a function, and it governs far more than just your lock screen.

Today's hook, Your phone's face unlock maps your face to exactly one answer: unlock or don't unlock. What would happen if the same face could produce two different answers? And what does this have to do with mathematics?

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Orient and recall

Meet the destination, commit to a gut answer, and gather the formulas and key terms you will lean on.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question

Recall, your gut answer first

+5 XP warm-up

Your phone's face unlock works because it has learned a rule: your face (the input) must produce exactly one answer, unlock or don't unlock. What do you think would happen if the same face could produce two different answers? And how is this idea connected to mathematics?

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Formula reference · this lesson

core notation

$$y = f(x)$$

$x$ = input (independent variable)

$f(x)$ = output (dependent variable)

$f$ = name of the function

Key insight: $f(x)$ is read as "$f$ of $x$", it describes a rule, not a multiplication.

What you'll master

Know

Key facts

the definition of a function and a relation
how to use function notation $f(x)$
the vertical line test

Understand

Concepts

why a function allows only one output per input
the difference between independent and dependent variables
how the vertical line test works graphically

Can do

Skills

evaluate functions for given inputs
apply the vertical line test to graphs
distinguish functions from relations

Key terms

Function

A relation where each input has exactly one output.

Domain

The set of all possible input values for a function.

Range

The set of all possible output values for a function.

Relation

A set of ordered pairs connecting inputs to outputs.

Vertical Line Test

A graph represents a function only if no vertical line crosses it more than once.

Ordered Pair

A pair $(x, y)$ showing an input and its corresponding output.

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What makes a relation a function

Pin down the one-input-one-output rule and learn to test a graph for it.

What is a function?

core concept · +3 XP at end

In mathematics, a relation is any set of ordered pairs that connects inputs to outputs. A function is a special kind of relation with one strict rule: each input must be connected to exactly one output.

Think of a function like a machine. You put a number in, the machine applies a rule, and exactly one number comes out. If the same input could produce two different outputs, the machine would be unpredictable, and it would no longer be a function.

Why does your phone unlock only with your face? Face ID stores a mathematical model of your face. When you look at the phone, it compares the new image (input) to the stored model. The rule must produce exactly one output: either "unlock" or "don't unlock." If the same face could produce both answers, the system would be useless.

The vertical line test

On a graph, we can test whether a relation is a function using the vertical line test:

If every vertical line intersects the graph at most once, the relation is a function.
If any vertical line intersects the graph more than once, the relation is not a function.

A graph is a function only if every vertical line meets it at most once.

A relation = any set of ordered pairs connecting inputs to outputs; A function = a relation where each input has exactly one output

Pause, copy the function definition (each input maps to exactly one output) and the vertical line test rule into your book.

Did you get this? True or false: a circle with equation $x^2 + y^2 = 25$ represents a function.

Quick check: Which of these is the best description of a function?

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Read and use function notation

Separate independent from dependent variables and read $f(x)$ correctly.

Variables & function notation

core concept

We just saw that a function requires exactly one output per input, and the vertical line test detects this on a graph. That raises a question: once we know something is a function, how do we write and evaluate it efficiently? This card answers it → introducing $f(x)$ notation and the independent/dependent variable distinction.

Every function has two kinds of variables:

Independent variable ($x$): The value you choose or control, the input.
Dependent variable ($f(x)$ or $y$): The value that results from applying the function's rule, the output.

$f(x)$ is read "$f$ of $x$", not "$f$ times $x$." The letter $f$ is simply the name of the function (like a machine's label). The parentheses tell you that $x$ is the input being fed into the function. This is one of the most common points of confusion in all of Year 11, so get it right from the start.

For example, if $f(x) = 2x + 3$, then:

$f$ is the name of the function
$x$ is the input (independent variable)
$2x + 3$ is the rule that produces the output (dependent variable)

We can also name functions with other letters: $g(x)$, $h(x)$, or even descriptive names like $C(t)$ for "cost as a function of time."

Independent variable: the input, what you choose (usually $x$); Dependent variable: the output, what depends on the input (usually $f(x)$ or $y$)

Pause, copy the independent/dependent variable definitions and the reading rule ($f(x)$ means "$f$ of $x$", not $f$ times $x$) into your book.

Fill the blanks: drag each token into the matching blank.

input output independent dependent

In $f(x)$, the variable $x$ is the ___ and the ___ variable. The value $f(x)$ is the ___ and the ___ variable.

Worked examples · reveal as you go

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Work two examples end to end

Follow a full evaluation and a full vertical line test, step by step.

Worked example 1 · evaluating a function +5 XP on full reveal

If $f(x) = x^2 - 3x + 5$, find the value of $f(2)$ and $f(-1)$.

$f(x) = x^2 - 3x + 5$

Identify the rule

$f(2) = (2)^2 - 3(2) + 5$

Substitute $x = 2$, use brackets to avoid sign errors

$f(2) = 4 - 6 + 5 = \mathbf{3}$

Calculate

$f(-1) = (-1)^2 - 3(-1) + 5$

Substitute $x = -1$, brackets are essential with negatives

$f(-1) = 1 + 3 + 5 = \mathbf{9}$ ✓

Calculate

Worked example 2 · the vertical line test +5 XP on full reveal

Determine whether the graph of $x = y^2$ represents a function.

Recall the vertical line test

If any vertical line intersects the graph more than once → not a function

Analyse the graph of $x = y^2$

This is a sideways parabola opening to the right. For any positive $x$-value, there are two $y$-values.

Apply the test

At $x = 4$: $y = 2$ and $y = -2$. A vertical line at $x = 4$ intersects the graph twice.

$x = y^2$ is NOT a function ✓

One input ($x = 4$) maps to two outputs ($y = 2$ and $y = -2$), violates the definition.

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Dodge the traps, then classify

Meet the three mistakes that cost marks, then sort relations and name variables yourself.

Common mistakes · the 3 traps that cost marks

Thinking $f(x)$ means $f$ multiplied by $x$

The notation $f(x)$ is function notation, not multiplication. $f$ is the name of the function and $x$ is the input. Treating it as multiplication leads to completely wrong answers in every function question.

✓ Fix: Always read $f(x)$ as "$f$ of $x$", the output of function $f$ when the input is $x$.

Using a horizontal line instead of a vertical line for the test

Some students use a horizontal line. The vertical line test checks whether one input maps to multiple outputs. A horizontal line test checks for one-to-one functions, that's a different concept entirely.

✓ Fix: Remember, vertical line = function test. If it crosses more than once, it's not a function.

Assuming all relations are functions

Circles, sideways parabolas, and many other graphs are relations but not functions. Every function is a relation, but not every relation is a function.

✓ Fix: Always check the "one output per input" rule before calling a relation a function.

Activity 1, Function or not a function?

For each relation below, decide whether it is a function or not. Explain your reasoning in terms of the "one input, one output" rule.

$y = 2x + 1$

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A circle with equation $x^2 + y^2 = 25$

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The set of ordered pairs: $(1, 2), (2, 3), (3, 4), (1, 5)$

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$y = |x|$

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Odd one out: Which of these is NOT a function? Select one.

Activity 2, Identify the variables

For each situation, identify the independent and dependent variables, then write the relationship using function notation.

A taxi ride costs $\$5$ as a flag fall plus $\$2$ per kilometre. The total cost $C$ depends on the distance $d$ in kilometres: $C(d) = 5 + 2d$.

Identify: (a) the independent variable; (b) the dependent variable; (c) write in function notation.

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The area of a square depends on its side length $s$: $A(s) = s^2$.

Identify: (a) the independent variable; (b) the dependent variable.

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A scientist records that the temperature $T$ of an object increases as it is exposed to sunlight for $t$ minutes. Write a function notation for this relationship.

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Drill it, then lock it in

Run the quick drill and copy the summary into your book.

Quick-fire practice · 5 reps +2 XP per reveal

Given $f(x) = 3x - 7$, find $f(4)$.

Given $g(x) = x^2 + 2x$, find $g(-3)$.

Does the graph of $y = x^3$ pass the vertical line test? Is it a function?

In $C(t) = 20 + 0.5t$, which is the independent variable and which is the dependent variable?

A relation has ordered pairs $(1,3), (2,5), (3,7), (2,9)$. Is it a function? Explain.

Revisit your thinking

Earlier you were asked: What would happen if the same face could produce two different answers? And how is this connected to mathematics?

If the same input (your face) could produce two different outputs (unlock and don't unlock), the system would be completely unreliable. In mathematics, this is exactly what distinguishes a function from a general relation: a function guarantees exactly one output for every input. This rule of unique outputs makes functions predictable, powerful, and essential for everything from phone security to engineering design.

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Multiple choice

Answer the drill bank and rate your confidence on each one.

Multiple choice

+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence, that tells the system what to drill next.

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Short answer

Write full responses, then check them against the model answers.

Short answer

UnderstandBand 33 marks

Q1. Explain the difference between a function and a relation. In your answer, describe how the vertical line test can be used to determine whether a graph represents a function. (3 marks)

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ApplyBand 33 marks

Q2. Consider the function $f(x) = 3x^2 - 2x + 4$. (a) Evaluate $f(2)$. (b) Evaluate $f(-1)$. Show all working. (3 marks)

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EvaluateBand 54 marks

Q3. A smartphone's face unlock system uses a mathematical rule to check whether an image matches the owner's face. The rule must always produce the same output for the same input. Evaluate whether the rule "$y$ equals the square root of $x$" ($y = \sqrt{x}$) would be suitable as the basis for a face unlock system. Justify your answer with reference to the definition of a function. (4 marks)

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📖 Comprehensive answers (click to reveal)

Multiple choice, drill bank

MC answers and feedback are shown inline as you complete each question. Use the retry button to attempt a fresh set.

Activity 1, Sort + Classify model answers

A. $y = 2x + 1$ Function. It is a straight line; any vertical line will intersect it exactly once. Every input $x$ gives exactly one output $y$.

B. $x^2 + y^2 = 25$ (circle) Not a function. A vertical line through the centre intersects the circle twice (e.g. at $x = 0$, $y = 5$ and $y = -5$). One input maps to two outputs.

C. $(1, 2), (2, 3), (3, 4), (1, 5)$ Not a function. The input $1$ appears twice with different outputs ($2$ and $5$), violating the one-input-one-output rule.

D. $y = |x|$ Function. The graph is a V-shape. Every vertical line intersects it at most once. Each $x$ produces exactly one $y$-value.

Activity 2, Identify the variables model answers

1. Independent = distance $d$ (km); Dependent = total cost $C$ in dollars. Notation: $C(d) = 5 + 2d$

2. Independent = side length $s$; Dependent = area $A$. Notation: $A(s) = s^2$

3. Independent = time $t$ (minutes); Dependent = temperature $T$ (°C). Notation: $T(t)$ (exact formula not required)

Short answer model answers

Q1 (3 marks): A relation is any set of ordered pairs that connects inputs to outputs [1]. A function is a special type of relation where each input is paired with exactly one output [1]. The vertical line test checks whether any vertical line intersects a graph more than once; if it does, the relation is not a function because one input would map to multiple outputs [1].

Q2 (3 marks):

(a) $f(2) = 3(2)^2 - 2(2) + 4 = 12 - 4 + 4 = \mathbf{12}$ ✓
(b) $f(-1) = 3(-1)^2 - 2(-1) + 4 = 3 + 2 + 4 = \mathbf{9}$ ✓
Award 1 mark each for correct evaluation with working shown.

Q3 (4 marks): The rule $y = \sqrt{x}$ is technically a function for $x \geq 0$ because the principal square root produces exactly one non-negative output for each non-negative input [1–2]. However, it is not suitable for a face unlock system: its domain is restricted to $x \geq 0$ [1], and far more critically, a face-recognition system requires a complex rule that can distinguish between millions of different faces, $y = \sqrt{x}$ is far too simple and maps all inputs to the same type of output regardless of their structure [1].

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Review and finish

Take the module quiz if you are ready, then mark the lesson complete.

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quiz

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Mark lesson as complete

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← Lesson 31 · Semicircles and Circles from a Graph Next: Function Notation & Evaluation →

Module overview · Maths Advanced · Module quiz