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.
Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.
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?
Type your initial response below — you will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
Wrong: A function can have two different outputs for the same input.
Right: By definition, a function has exactly one output for each input in its domain.
📚 Core Content
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.
On a graph, we can test whether a relation is a function using the vertical line test:
Every function has two kinds of variables:
For example, if $f(x) = 2x + 3$, then:
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."
🧮 Worked Examples
🧪 Activities
$y = 2x + 1$
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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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1 A taxi charges a $5 flag fall plus $2 per kilometre travelled.
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2 The area of a square depends on the length of its side.
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3 The temperature inside a car depends on how long it has been sitting in the sun.
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Earlier you were asked: What do you think would happen if the same face could produce two different answers? And how is this idea connected to mathematics?
If the same input (your face) could produce two different outputs (unlock and don't unlock), the system would be 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.
Now revisit your initial response. What did you get right? What has changed in your thinking?
Look back at your initial response in your book. Annotate it with what you now understand differently.
Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?
5 random questions from a replayable lesson bank — feedback shown immediately
✍️ Short Answer
8. 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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9. 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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Answer in your workbook.
10. 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 mathematical 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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Answer in your workbook.
A. $y = 2x + 1$ — Function. It is a straight line, so 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 the graph at most once. Each $x$ produces exactly one $y$-value.
1. Independent = distance travelled (km); Dependent = total cost ($); Notation: $C(d) = 5 + 2d$
2. Independent = side length; Dependent = area; Notation: $A(s) = s^2$
3. Independent = time in sun (minutes); Dependent = temperature ($^\circ$C); Notation: $T(t)$ (exact formula not required)
1. B — A function is a relation where each input has exactly one output.
2. B — $f(3)$ is the output of the function $f$ when the input is 3.
3. C — The vertical line test determines whether a relation is a function.
4. A — $f(-1) = 2(-1) + 5 = 3$ (note: this distractor depends on the exact questions drawn).
5. C — $x = y^2$ is not a function because one $x$-value can produce two $y$-values.
Q8 (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].
Q9 (3 marks):
(a) $f(2) = 3(2)^2 - 2(2) + 4 = 12 - 4 + 4 = 12$ (b) $f(-1) = 3(-1)^2 - 2(-1) + 4 = 3 + 2 + 4 = 9$Award 1 mark for each correct evaluation and 1 mark for showing working.
Q10 (4 marks): The rule $y = \sqrt{x}$ is suitable as a function for $x \geq 0$ because for every non-negative input $x$, the principal square root produces exactly one output $y$ [1–2]. However, it would not be suitable for a face unlock system because the domain is restricted to $x \geq 0$ [1]. More importantly, a face unlock system requires a much more complex rule that distinguishes between millions of different faces, whereas $y = \sqrt{x}$ is far too simple and would assign the same output to completely different inputs [1]. Therefore, while it satisfies the definition of a function on its domain, it is not practically suitable.
Climb platforms using your knowledge of functions, relations and mapping notation. Pool: lesson 1.
Tick when you've finished all activities and checked your answers.