H2 Maths Functions Formula Sheet | Domain, Range & Inverses

Study guideUpdated 21 Aug 2026

H2 Maths functions formula sheet: domain and range rules, the one-one test, conditions for inverses to exist, domain and range swap under inversion, the reflection property of i...

Q: What does H2 Maths Notes (JC 1-2): 1.1) Functions cover?
A: Domains, ranges, inverse checks, and composite workflows for Section A Topic 1.1 of the 2026 H2 Maths syllabus.
Download: Get the H2 Maths Functions formula sheet (PDF) for quick revision, or the complete notes (PDF) for the full walkthrough.
Before you revise
Keep a running list of domain restrictions (denominators, even roots, logarithms). Test every claim with quick substitutions or GC trace-functions questions are marked on precision.
  • A function needs valid inputs and outputs: State the domain.
  • Inverses need one-to-one behaviour: Restrict the domain before finding the inverse.
  • Composite functions carry restrictions through each layer: Check the inner function's range against the outer function's domain.

Concrete example: For f(x)=3x−5f(x)=\sqrt{3x-5}, the input must satisfy 3x−5≥03x-5 \geq 0

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 1.1 expectations are within Section A Pure Mathematics, which is assessed in Paper 1 (100 marks) and Paper 2 Section A (40 marks).

Formulas at a glance

Every key result, condition, and rule from this topic, on one screen. None of these definitions appear in MF27, so commit them to memory. Worked examples for each appear in the sections below.

Definitions and domain rules

ConceptRule
Functionf:A→Bf: A \to B assigns each x∈Dfx \in \mathrm{D}_f to exactly one output f(x)f(x)

One-one (injectivity) test

MethodConditionWhat to write
AlgebraicAssume f(x1)=f(x2)f(x_1) = f(x_2) and deduce x1=x2x_1 = x_2

Inverse functions

ConceptRule
Existence conditionf−1f^{-1} exists if and only if ff is one-one on Df\mathrm{D}_f.

Composite functions

ConceptRule
Existence conditionfgfg exists if and only if Rg⊆Df\mathrm{R}_g \subseteq \mathrm{D}_f.

Function question route map

Use the first cue in the question to choose the workflow before doing algebra. Most errors in functions questions come from finding a formula first and checking the domain too late.

Question cueWhat to decide firstWorking route
"State the domain"Which inputs are illegal?Check denominators, even roots, logarithms, then combine the restrictions.
"Find the range"Which outputs can the function actually reach?Use graph shape, monotonicity, completed square form, or a boundary-value table.
"Find the inverse"Is the given domain one-to-one?Restrict or verify the domain, rearrange, then swap the domain and range.
"Find a composite"Does the inner output fit the outer input?Find the inner function's range first, then apply the outer function only where valid.
"Solve an equation involving functions"Which layer can be undone first?Let the inner expression be a new variable, solve the outer equation, then back-substitute.

Trap check: If a question gives a restricted domain, use that restricted domain for the range and inverse. Do not silently switch back to the natural domain halfway through the solution.

Core Concepts

  • A function f:A→B f: A \to B assigns each x∈A x \in A to one output f(x)∈B f(x) \in B ; A A

Domain and Range Workflow

  1. Determine the natural domain by eliminating forbidden inputs:
    • Even roots: require the radicand ≥0 \geq 0 .
    • Logarithms: argument strictly >0 > 0 .
    • Rational functions: denominator ≠0 \neq 0 .
    • Composite functions: propagate restrictions through every stage.
  2. State the range by analysing turning points, asymptotes, or monotonic intervals.
  3. Restrict the domain if necessary to make f f

Example -- Square root domain

For f(x)=3x−5 f(x) = \sqrt{3x - 5} , require 3x−5≥0 3x - 5 \geq 0

Domain-range transfer checkpoint

For inverse questions, write the four sets in this order before rearranging the formula: original domain, original range, inverse domain, inverse range. This prevents a common mistake where the natural domain is used after the question has already restricted the function.

Item to stateComes fromWhat to check
Domain of ffThe question or the natural-domain restrictionsDenominators, even roots, logarithms, and any given interval.
Range of ffOutputs reached only on that domainBoundary values, turning points, asymptotes, and monotonic branch chosen.
Domain of f−1f^{-1}

Worked check: suppose f(x)=x2−4x+1f(x)=x^2-4x+1 and the question restricts x≥2x \geq 2. Completing the square gives f(x)=(x−2)2−3f(x)=(x-2)^2-3

Misconception check: "swap xx and yy" is not the first step. First decide which branch of the original function is being inverted, then swap the restricted domain and range.

Injectivity proof checkpoint

Before finding an inverse, decide how you will justify that the function is one-to-one on the stated domain. Choose the shortest proof that matches the information given.

Given informationFast proof routeWhat to write
A simple algebraic formulaStart from f(a)=f(b)f(a)=f(b).Show the equation forces a=ba=b, so the function is one-to-one.
A differentiable function on an intervalCheck the sign of f′(x)f'(x)

Worked check: for f(x)=x2−6x+5f(x)=x^2-6x+5, completing the square gives f(x)=(x−3)2−4f(x)=(x-3)^2-4

Misconception check: "the graph has a turning point" does not automatically mean the inverse is impossible. It means you must restrict to one side of the turning point before inverting.

Inverse Functions

  • Use algebraic manipulation to isolate x x in y=f(x) y = f(x) ; then interchange x x and y y to express f−1(x) f^{-1}(x)
The graph of a function y = f(x) and its inverse y = f⁻¹(x), which is the mirror image of f in the line y = x. Reflecting any point (a, b) on f gives the point (b, a) on the inverse.xyOy = xy = f(x)y = f⁻¹(x)
The graph of an inverse function y = f⁻¹(x) is the reflection of y = f(x) in the line y = x: reflecting a point (a, b) on f gives (b, a) on the inverse. The inverse exists as a function only when f is one-one, so its domain is the range of f.

Example -- Quadratic inverse

Consider f(x)=2x2−3x−1 f(x) = 2x^2 - 3x - 1 .

  1. Stationary point from f′(x)=4x−3=0 f'(x) = 4x - 3 = 0 gives x=34 x = \tfrac{3}{4}

Composite Functions

  • Record intermediate ranges meticulously: if g:A→B g: A \to B and f:C→D f: C \to D , composition requires g(A)⊆C g(A) \subseteq C

Composite-domain checkpoint

For a composite function, the inner function is checked first, but the final domain must satisfy both the inner function and the outer function.

TaskFirst checkThen checkCommon trap
Find (f∘g)(x) (f \circ g)(x) Inputs must be valid for g g .Outputs from g g must be valid inputs for f f

Worked check: let f(x)=x−2 f(x)=\sqrt{x-2} and g(x)=x2−1 g(x)=x^2-1

Misconception check: simplifying the algebra can hide the route taken by the composite. Always record the inner-output restriction before finalising the domain.

Example -- Composition with logarithms

Let f(x)=ln⁡(x−1) f(x) = \ln(x - 1) and g(x)=x2+4 g(x) = x^2 + 4 .

  1. Natural domain of g g is all reals, range [4,∞) [4, \infty) .
  2. f f requires input >1 > 1 , so g(x)>1 g(x) > 1

Modelling with Functions

  • When modelling real data, specify domain in context (e.g. weeks, temperatures).
  • Piecewise definitions handle regime changes: f(x)={2x+3,x<1x2,x≥1 f(x) = \begin{cases} 2x + 3, & x \lt 1 \cr x^2, & x \geq 1 \end{cases}

Calculator Workflow

  • Use TABLE mode to scan domain issues quickly (undefined entries highlight restrictions).
  • CALC > min / max helps verify monotonic intervals before restricting domains.
  • Document the command sequence in worked solutions, e.g. “GC: ISCT between y1=2x2−3x−1 y_1 = 2x^2 - 3x - 1 and y2=k y_2 = k

Practice Quiz

Test your command of domains, inverses, and composition pitfalls before moving to graph work.

Quick Revision Checklist

  • Identify domains and ranges, quoting restrictions explicitly.
  • Prove injectivity via algebra or derivative sign to justify inverses.
  • Execute compositions and inverses while checking compatibility of domains.
  • Translate contextual descriptions into precise function definitions.

Next steps: Keep working through the H2 Maths notes hub and jump into Topic 1.2 - Graphs & Transformations for full sketch workflows.

Want weekly guided practice on Functions? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.

Common exam mistakes

  • Giving the range of ff as the domain of f−1f^{-1} without checking: The domain of f−1f^{-1} equals the range of ff

Frequently asked questions

Is there a formula sheet for H2 Maths functions?
Yes - the "Formulas at a glance" section near the top of this page collects every key result you need: the definition of a function, natural domain rules, the one-one test (algebraic, derivative, and horizontal-line methods), the existence condition for inverses, the domain-range swap under inversion, the reflection property of inverse graphs, the composite-existence condition Rg⊆Df\mathrm{R}_g \subseteq \mathrm{D}_f, the order of composition, and the domain of a composite. None of these appear in MF27, so commit them to memory.

Is Topic 1.1 (Functions) in Paper 1 or Paper 2?
Topic 1.1 is Pure Mathematics and can appear in Paper 1 (100 marks) or Paper 2 Section A (40 marks). Functions questions are often in the early parts of structured questions and test precision with domain notation.

Do I need to prove injectivity algebraically, or is a GC sketch sufficient?
The GC can support your reasoning but is not a substitute for an algebraic or derivative-based proof. To prove ff is one-to-one, either show f(x1)=f(x2)⇒x1=x2f(x_1) = f(x_2) \Rightarrow x_1 = x_2

What is the difference between "domain" and "natural domain"?
The natural domain is the largest set of real numbers for which f(x)f(x) is defined (no division by zero, no negative square roots, etc.). A restricted domain is a subset chosen to make ff injective (for finding an inverse). Always check whether the question specifies a domain or asks you to state the natural domain.

Other H2 Maths formula sheets

Revising more than one topic? Grab the matching one-page formula sheet:

For the official SEAB reference booklet, see the H2 Maths MF27 formula list.

Sources

  • SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 1.1 Functions (definitions, domain/range, composition, inverse functions) under Section A Pure Mathematics: SEAB H2 Mathematics syllabus PDF

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Marcus Pang
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Marcus Pang·Managing Director (Maths)

Sources

  1. SEAB H2 Mathematics (9758) Syllabus 2026