Pure Mathematics 3

Topic Brief: Functions Composition and Inverses

Provenance: compiled 2026-09-29, from research/veridian/WMA13-verified-facts.md (this course's primary, provenance-checked WMA13 source) and codex-deep-dive-functions-composition-and-inverses.md (this folder — the pass that independently opened six primary Pearson documents for this exact topic and re-verified every anchor below against a real mark scheme), cross-checked against signal-frequency-and-boundaries.md, forge-exam-technique-guide.md and codex-formula-and-method-bank.md so this brief does not restate their content. No new primary-source material was pulled for this pass — this is a synthesis of what those documents already establish, reorganised as a short "what to expect" reference for this one spec point.

#SIGNAL Topic Master Brief: Functions — Composition and Inverses (WMA13, spec 1.2)

#What this document is

The lesson (functions-composition-and-inverses.ts) teaches the derivations and two worked chains. The CODEX deep-dive for this topic surveys the real algebraic families this sub-topic is examined through and builds a full trap taxonomy across nine series. This document does neither — it is the short version: which real series tested this topic, at what question and tariff, in what recurring shape, and what to expect if it appears again. Full workings live in the CODEX deep-dive; this brief only cites and classifies.

#Frequency

Seen in 9 of the 13 WMA13 series checked so far (count from signal-frequency-and-boundaries.md, not recomputed here): Oct 2021, Jan 2022, Jun 2022, Oct 2022, Jun 2023, Jan 2024, Jan 2023, Oct 2024, Jun 2026. Of those nine, eight have real question text and mark-scheme detail independently re-verified against a primary Pearson PDF (all but Oct 2021, whose archived PDF could not be re-opened for the CODEX pass — that one citation rests on an examiner-report quote already verified elsewhere in this course's research, not on a fresh re-read of the question paper). Not confirmed present in the other four series checked (Oct 2020, Jun 2024, Oct 2023, Jan 2025) on the evidence gathered so far.

#The anchors, cited by series and question

Series	Q	Real content
Jan 2023	Q1 (6 marks total)	Range of a quadratic; composition $fg(1.5)$ ; inverse of a linear-over-linear ("Möbius") function $g(x)=\frac{3}{2x+1}$ , with the domain a separate, independent mark from the algebra.
Jan 2022	Q6	Inverse of a Möbius $f(x)=\frac{5x-3}{x-4}$ ; a self-composition $ff(x)$ collapsing to another Möbius function, with its range read from the domain's own two boundary limits, not calculus.
Jun 2022	Q2	Inverse of a Möbius $f(x)=\frac{5-x}{3x+2}$ ; a mixed-argument equation, $f(\frac1a)=g(a+3)$ , pairing two different functions' arguments.
Oct 2022	Q2	Inverse of a Möbius $f(x)=5-\frac{4}{3x+2}$ ; the named real errors are deriving the inverse's domain "from scratch" and confusing $f^{-1}$ with the derivative $f'$ .
Jun 2023	Q4	A purely graphical part — sketch $y=f^{-1}(x)$ directly from a sketch of $y=f(x)$ , no algebra — followed by finding where the two curves meet, which the real mark scheme prices as solvable by a $y=x$ shortcut or by a much harder literal route.
Jan 2024	Q4	Inverse of a logarithmic function $g(x)=3+2\ln x$ ; solving $gf(a)=5$ , with the real mark scheme printing two equally credited routes.
Oct 2024	Q6 (2+3+3 = 8 marks)	Inverse of a Möbius $f(x)=6-\frac{21}{2x+3}$ ; a self-composition $gg(x)=126$ , whose mark scheme explicitly zero-marks the answer produced by undoing the rule once instead of twice.
Jun 2026	Q2 (9 marks total)	Composition at an algebraic argument, $fg(a^2-5)=g(59)$ , needing a stated sign constraint to collapse a square root before an equation can even be set up.
Oct 2021	Q1(b)	Domain of the inverse omitted — attested by examiner-report quote only; the question's own full text was not independently re-opened for this pass.

#Mark-tariff pattern

This topic's tariff runs from a compact 6-mark opener when it stands alone (Jan 2023 Q1) up to 8–9 marks when the same question bundles a second composition, a self-composition, or an algebraic-argument collapse alongside the ordinary domain/range/inverse work (Oct 2024 Q6: 2+3+3=8; Summer 2026 Q2: 1+3+2+3=9). No located anchor tests this topic for a token 1–2 marks — every real instance found carries substantive weight, usually spread across two or three sub-parts of one question rather than concentrated in a single part.

#Recurring context and angle type

Across the nine anchors, four shapes recur:

Bundled, early placement. Every anchor is a multi-part question combining some mix of domain/range, composition, and inverse-with-domain sub-parts, rather than a single-skill standalone question.
A linear-over-linear ("Möbius") rational function, not a quadratic, is the more common real vehicle for the inverse sub-part — five of the nine anchors (Jan 2023, Jan 2022, Jun 2022, Oct 2022, Oct 2024) invert a function of this shape, against the taught lesson's own two worked chains, both restricted quadratics.
Self-composition — a function composed with itself ( $gg$ or $ff$ ), not two different functions — is a distinct, recurring mechanism, confirmed in two series (Jan 2022 Q6, Oct 2024 Q6), each carrying its own "undo the rule twice, not once" error.
Domain/range notation direction is examined from both sides: a range stated in $x$ -notation loses the mark (Jan 2023 Q1(a)), and a domain stated in output notation loses the mark too (Jan 2024 Q4(c)) — confirmed on two separate real mark schemes, not one topic's worth of the same slip.

#If this topic appears — what to expect

Expect this topic early in the paper as a multi-part opener, not a single-skill question. The function to be inverted is more likely a linear-over-linear fraction than a quadratic on the real evidence gathered so far, so the cross-multiply route is worth being at least as fluent in as the complete-the-square route. State the domain of an inverse as its own separate line, in $x$ -notation, and a range in $y$ / $f(x)$ -notation — both directions of this mix-up are real, mark-scheme-confirmed losses. If a question composes a function with itself, remember that "solve $gg(x)=k$ " requires undoing the rule twice; a plausible-looking answer from undoing it once has been explicitly zero-marked on a real scheme. If asked to sketch or locate an intersection of $f$ and $f^{-1}$ , check for a $y=x$ shortcut before committing to algebra — the one real instance of this angle shows the literal route leading to a much harder equation.

#Honest gaps

Not confirmed present in four of the thirteen series checked (Oct 2020, Jan 2025, Oct 2023, Jun 2024) — this is absence of evidence in this course's research, not evidence the topic was absent from those papers. The Oct 2021 anchor rests on an examiner-report quote already verified elsewhere in this course's research; its underlying PDF could not be independently re-opened this pass to confirm the full question text. No likelihood, percentage, or grade-boundary claim is made anywhere above; every count is a "seen in N of M series checked" citation traceable to an existing document in this folder.

#Sources

research/veridian/WMA13-verified-facts.md — §1 (spec wording), §5 (1.2 Functions, for the Jan 2023 and Summer 2026 anchors' verbatim detail), §7, §9 (lesson-audit log).
codex-deep-dive-functions-composition-and-inverses.md (this folder) — the full mark-by-mark detail, micro-examples, and trap taxonomy for every anchor above; consult it for workings, not this brief.
signal-frequency-and-boundaries.md (this folder) — the "9 of 13" frequency count, reused rather than recomputed.
codex-formula-and-method-bank.md, forge-exam-technique-guide.md (this folder) — the compressed M/A/B skeletons and the Oct 2024 tariff split.
content/veridian/curriculum/wma13/functions-composition-and-inverses.ts — the taught derivations and worked chains, not reproduced here.