A Level Mathematics: Pure Mathematics 1 - Functions (Cambridge 9709)

Subtopic 1.2 Functions follows 1.1 Quadratics in Pure Mathematics 1 (Paper 1) of Cambridge International AS & A Level Mathematics 9709. Where the Pure Mathematics 1 overview summarises all eight sub-topics briefly, this guide goes deep into 1.2 specifically – the vocabulary and techniques of functions that recur throughout the rest of the Pure Mathematics content, from Paper 2’s logarithmic and exponential functions to Paper 3’s more advanced function work.

The core vocabulary

You need to understand five terms precisely: function, domain, range, one-one function, and inverse function, plus composition of functions. A function maps every value in its domain to exactly one value in its range. A one-one function is one where no two different domain values map to the same range value – this matters because only a one-one function has a genuine inverse function.

Finding range and composing functions

You need to identify the range of a given function in simple cases, and find the composition of two given functions. The syllabus gives a worked style of example: for f: x ↦ 1/x for x ≥ 1 and g: x ↦ x + 1, you should be able to find the range of the composite function gf.

The critical rule for composition: a composite function gf can only be formed when the range of f lies within the domain of g. This condition is easy to overlook – always check it explicitly before attempting to build a composite function, since a question may be testing precisely whether you notice the composition is invalid as stated.

Determining one-one and finding inverses

You need to be able to determine whether a given function is one-one, and find the inverse of a one-one function in simple cases. A common technique for checking one-one-ness is considering whether a horizontal line could cross the graph of the function more than once (if it can, the function is not one-one over that domain).

To find an inverse function algebraically: write y = f(x), rearrange to make x the subject, then swap x and y (or relabel) to express the inverse in terms of x.

The graphical relationship between a function and its inverse

You need to be able to illustrate, in graphical terms, the relation between a one-one function and its inverse. The key fact to sketch correctly: the graph of the inverse function is the reflection of the graph of the original function in the line y = x. Any sketch answering this kind of question should explicitly show this mirror line, since the syllabus notes that sketches should include an indication of it.

Graph transformations

The final part of 1.2 covers four standard transformations of the graph of y = f(x), which you need to understand and use, including simple combinations of them:

Transformation	Effect
y = f(x) + a	Vertical translation by a
y = f(x + a)	Horizontal translation by −a
y = a·f(x)	Vertical stretch, scale factor a
y = f(ax)	Horizontal stretch, scale factor 1/a

You need to use the terms translation, reflection and stretch correctly when describing these transformations, and be ready to apply them to algebraic functions, trigonometric functions, or other graphs described only by their general features rather than a specific equation.

Worked-style examples to practise

Two example patterns are worth drilling until they are automatic, because they cover most of what a 1.2 Functions question asks.

Composition and its condition. Suppose f: x ↦ 1/x for x ≥ 1 and g: x ↦ x + 1 for x ∈ ℝ. The range of f (for x ≥ 1) is 0 < f(x) ≤ 1. Since g accepts all real numbers as its domain, the range of f lies entirely within the domain of g, so gf exists and gf(x) = 1/x + 1. Before writing this down in an exam, state the domain-and-range check explicitly as a line of working – it is often worth a mark on its own, independent of the final expression.

Inverse of a restricted function. Suppose f: x ↦ (x − 2)² + 3 for x ≥ 2. Because the domain is restricted to x ≥ 2, f is one-one (the horizontal-line test only fails for the full parabola, not the restricted right-hand branch), so an inverse exists. Write y = (x − 2)² + 3, rearrange to x = 2 + √(y − 3), then swap variables to give f⁻¹(x) = 2 + √(x − 3) for x ≥ 3. Note how the domain restriction on f becomes the range restriction on f⁻¹, and vice versa – this domain/range swap between a function and its inverse is a detail examiners check for specifically.

How 1.2 connects to the rest of the syllabus

Functions is genuinely foundational, not a self-contained topic you revise once and set aside. The domain/range/composition vocabulary recurs whenever a later Pure Mathematics topic introduces a new family of functions (logarithmic and exponential in Paper 2, further algebraic and trigonometric functions in Paper 3), and the translation/stretch/reflection transformations apply directly to sketching trigonometric graphs in 1.5 and any curve-sketching question across the whole qualification.

How to approach it

Practise the composition condition (range of the inner function within the domain of the outer function) as a habit, not an afterthought – write it down explicitly before combining two functions, since exam questions specifically probe whether candidates check it. For inverses, practise the full algebraic method (isolate x, then swap variables) until it is automatic, and always sketch the y = x mirror line when a question asks for a graphical relationship between a function and its inverse, since marks are frequently allocated specifically for showing this line. For transformations, build a mental habit of applying them one at a time when a question combines two or more (for example, y = 2f(x − 1) is a horizontal translation followed by a vertical stretch), rather than trying to visualise the combined effect in a single step.

Official syllabus

Cambridge International, Cambridge International AS & A Level Mathematics (9709) syllabus for examination in 2026 and 2027: official syllabus PDF, Subject content, section 1.2 “Functions” (Pure Mathematics 1, Paper 1). Verified 2026-09-02.