Deep Dive: Functions Composition and Inverses

Provenance (2026-09-29): read/checked for this pass — research/veridian/WMA13-verified-facts.md (grepped and read for "composit", "inverse", "domain", "1.2", the spec table in §1, and the two dated audit-log entries for this lesson in §9); content/veridian/curriculum/wma13/functions-composition-and-inverses.ts (the full 939-line lesson, read start to end — its two worked chains, six-item trap taxonomy, beyond-spec self-inverse note, and six MCQs — to identify what already exists and avoid duplicating it); content/veridian/technique-hub/wma13/codex-formula-and-method-bank.md, forge-exam-technique-guide.md and signal-frequency-and-boundaries.md (grepped for 1.2 coverage — all three give compressed method skeletons or frequency counts, not a theory or synthesis treatment); content/veridian/question-bank/wma13.ts (grepped for fg( / gf( / inverse — no dedicated 1.2 item bank beyond the lesson's own six MCQs, so no duplication risk). Six real primary Pearson documents were opened fresh for this pass and independently verified beyond what WMA13-verified-facts.md already recorded — January 2022 Q6, June 2022 Q2, October 2022 Q2, June 2023 Q4, January 2024 Q4, all pdftotext -layout against the QP/MS/ER PDFs at ~/Documents/SOL/Papers/IAL_Paper3_Complete/IAL_Materials/Mathematics/<series>/, with October 2022 Q2's mangled modulus sign confirmed by rendering the actual question-paper page to a PNG (pdftoppm -r 250) and reading it directly, the same discipline WMA13-verified-facts.md itself documents for stubborn typesetting. October 2024 Q6, already logged as a citation-currency gap for the lesson's exponential neighbour, was independently re-opened here for its own functions content, which no existing WMA13 document had yet examined. Every numeric result quoted below (inverses, composite values, ranges, the self-inverse condition) was independently recomputed with sympy/Python before being written in, not copied from a mark scheme and trusted. An attempted seventh check, October 2021 Q1(b), found the archived PDFs at that path are corrupted (file reports HTML, not PDF) — that citation is carried over from WMA13-verified-facts.md's own already-verified quote rather than re-derived here, and is flagged as such at the point it is used.

#CODEX Deep-Dive: Functions — Composition and Inverses (WMA13, spec 1.2)

#What this document is for

functions-composition-and-inverses.ts already derives the composite-domain mechanism and the reflection-in-y=x mechanism for inverses from first principles, drills both through two full worked chains (a restricted quadratic each time), and carries six real, evidence-grounded traps anchored on January 2023 Q1(c) and Summer 2026 Q2(d). codex-formula-and-method-bank.md compresses those same two anchors into mark-scheme skeletons. signal-frequency-and-boundaries.md states the topic is tested in 9 of the series this course's research has opened — a frequency count with no detail behind it. None of the three lays out the range of algebraic families this sub-topic is actually examined through beyond the restricted quadratic, and none has opened the six primary sources this document opens below. This document does three things the existing files do not: (1) extends the theory one step further than the spec strictly requires — a general, self-verified condition for when a function is its own inverse, checked against every real linear-fractional function this research has actually located; (2) surveys the real algebraic families WMA13 has tested this sub-topic through, most of them absent from the existing lesson, each with a short, self-verified micro-example; (3) synthesises every real exam angle located across all nine confirmed series into one trap taxonomy, several of which are genuinely new findings from the six sources opened for this pass.

Spec 1.2's own wording (verified-facts §1, transcribed from the spec PDF): "Definition of a function. Domain and range of functions. Composition of functions. Inverse functions and their graphs," with guidance adding that students should know "that fg means 'do g first, then f'" and that "if f⁻¹ exists, then f⁻¹f(x) = ff⁻¹(x) = x."

#1. Theory the lesson already derives — stated briefly, not reproduced

A function is one-one or many-one on its stated domain; an inverse exists only where it is one-one, because a many-one map would force $f^{-1}$ to send one output back to two different inputs, which no function can do. Restricting a many-one function's domain to a single branch (the lesson's own $x \geq 2$ quadratics) is the standard fix. $fg$ means "apply $g$, then apply the result to $f$"; a composite's own domain is whatever makes $g(x)$ a valid input to $f$, which can translate a restriction on $f$ back into a different-looking restriction on $x$. Finding $f^{-1}$ is swapping $x$ and $y$ in $y=f(x)$ and rearranging — geometrically a reflection in $y=x$ — which is exactly why the domain of $f^{-1}$ equals the range of $f$, and vice versa. All of this is derived, diagrammed and drilled in full inside the lesson file; nothing above is new, and none of it is re-derived here.

#2. One step beyond the spec: when is a function its own inverse?

The lesson's own beyond-spec note names three self-inverse examples — $\frac{1}{x}$, $-x$, $c-x$ — and says correctly that the spec never requires knowing why. Every real linear-fractional function this research has located (§3.2 below) has the shape $f(x)=\frac{ax+b}{cx+d}$, so it is worth actually deriving the general condition, both because it is short and because it lets every one of those real functions be checked against it rather than eyeballed.

Composing $f$ with itself symbolically and subtracting $x$: $f(f(x)) - x = \dfrac{-(a+d)\big(cx^2+(d-a)x-b\big)}{c(ax+b)+d(cx+d)}$ (independently expanded and factored with sympy). The quadratic factor in the numerator is not identically zero for a genuine linear-fractional function (it needs $c \neq 0$ and $ad-bc\neq0$ to be a real Möbius map in the first place), so $f(f(x))=x$ for every $x$ in the domain if and only if $a+d=0$ — the two "diagonal" coefficients are negatives of each other. Checked directly: $f(x)=\frac{1}{x}$ has $a=0,d=0$, so $a+d=0$ ✓; $f(x)=c-x=\frac{-x+c}{0x+1}$ has $a=-1,d=1$, so $a+d=0$ ✓ — both match, confirming the derivation reproduces the lesson's own two Möbius-shaped examples.

Checking the condition against every real linear-fractional $f$ or $g$ this document's own research has located (table built from §3.2 below, each entry independently re-derived there):

None of the five real linear-fractional functions this research has located is self-inverse — every one has $a+d\neq0$, confirmed exactly (not approximately) with sympy in every case. That is a genuine, if negative, finding: on the evidence actually available, guessing $f^{-1}=f$ as a shortcut on a real WMA13 Möbius-type question would be wrong every single time it has been checked. It is stated here precisely because a document that only ever shows examples where a shortcut works would misrepresent how rarely, on this evidence, it actually does.

#3. The real algebraic families this sub-topic has actually been tested through

#3.1 Restricted quadratic → square-root inverse

Already the lesson's own two worked chains ($f(x)=x^2-4x+7,\,x\geq2$ and $f(x)=(x+1)^2-6,\,x\leq-1$) and its fci-marked-solution. Not repeated here.

#3.2 Linear-over-linear ("Möbius") rational → cross-multiply inverse

This is, on the evidence gathered across this document and the existing lesson combined, the single most common real shape for an $f^{-1}$ question on WMA13 — five distinct real anchors located, each independently re-solved with sympy and matched exactly against its real mark scheme:

Two genuinely different real domain shapes appear in this one family: a restriction to a half-line ($x\geqslant0$ or $x>4$, giving $f^{-1}$ a bounded or half-bounded domain) and an excluded single point ($x\neq-\tfrac23$, giving $f^{-1}$'s domain an excluded point too, not an inequality at all) — the lesson's own two worked chains only ever use the half-line case. Every one of the six algebra routes shown in the real mark schemes for these five questions is the same two-step method (isolate the fraction, cross-multiply, collect the swapped variable's terms on one side, factor it out) regardless of which of the two domain shapes applies — the domain shape changes what has to be stated at the end, not how the inverse itself is found.

Micro-example, fresh numbers, self-verified. $h(x)=\dfrac{2}{x+3}-1,\,x>-3$. Range: as $x\to-3^+$, $h\to+\infty$; as $x\to\infty$, $h\to-1^+$; $h$ is strictly decreasing (its only term with $x$ in it is a positive multiple of $\frac{1}{x+3}$, decreasing), so range is $h(x)>-1$. Swap and rearrange: $x=\frac{2}{y+3}-1 \Rightarrow x+1=\frac{2}{y+3} \Rightarrow y+3=\frac{2}{x+1} \Rightarrow y=\frac{2}{x+1}-3$, i.e. $h^{-1}(x)=\dfrac{2}{x+1}-3,\,x>-1$ — independently confirmed with sympy (solve(Eq(y,h),x) returns the same expression) and by substitution ($h(1)=-0.5$; $h^{-1}(-0.5)=1$ ✓).

#3.3 Logarithmic → exponentiate-both-sides inverse

Not present anywhere in the existing lesson, and a genuinely different rearrangement mechanism from either family above: rearranging isolates the $\ln$ term, then both sides are exponentiated rather than cross-multiplied or square-rooted. Real anchor, January 2024 Q4(c), independently verified against the primary mark scheme (WMA13_01_2401_MS): $g(x)=3+2\ln x,\,x>1 \Rightarrow g^{-1}(x)=e^{\frac{x-3}{2}},\,x>3$ (M1 A1 B1). Checked: $g(1)=3+2\ln1=3$, and $g\to\infty$ as $x\to\infty$, so range of $g$ is $(3,\infty)$, matching $g^{-1}$'s stated domain $x>3$ exactly.

The real examiner report for this question, quoted verbatim, records two errors specific to this family that a square-root or cross-multiply inverse cannot produce: "rearranging to $y = 3 + \ln x$ instead of $y = 3 + 2\ln x$" (the coefficient on the $\ln$ term dropped before exponentiating, not after) and, on the domain, "the fully correct domain of $x \geqslant 3$ was not often seen, with some candidates making statements about $g^{-1}(x)$ instead of $x$" — a domain stated in output notation ($g^{-1}(x)\geqslant3$, describing an output) rather than input notation ($x\geqslant3$), the exact mirror image of the already-documented range-must-be-$y$-notation trap the lesson's own trap taxonomy carries for January 2023 Q1(a). Both directions of this notation confusion are now independently confirmed on real mark schemes: a range stated in $x$-notation loses the mark (Jan 2023 Q1(a)), and a domain stated in $g^{-1}(x)$-notation loses the mark (Jan 2024 Q4(c)) — the same underlying rule, input-notation for domains and output-notation for ranges, catching students from both sides.

Micro-example, fresh numbers, self-verified. $k(x)=5-3\ln(2x),\,x>0$. Rearrange: $y=5-3\ln(2x)\Rightarrow \ln(2x)=\frac{5-y}{3}\Rightarrow 2x=e^{\frac{5-y}{3}}\Rightarrow x=\frac12e^{\frac{5-y}{3}}$, so $k^{-1}(x)=\frac12e^{\frac{5-x}{3}}$ — confirmed with sympy, and $k(1)=5-3\ln2\approx2.9207$, $k^{-1}(2.9207)\approx1.0000$ ✓.

#3.4 Self-composition — two distinct real mechanisms, neither in the existing lesson

Every composition in the existing lesson is $fg$ or $gf$ (two different functions). Two real WMA13 questions instead compose a function with itself, and the two real mechanisms this produces are structurally different from each other as well as from ordinary composition:

(a) Solving $gg(x)=k$ by "undoing twice." Real anchor, October 2024 Q6(c), verified against the primary mark scheme (WMA13_01_2410_MS): $g(x)=x^2+5,\,x\in\mathbb{R}$, solve $gg(x)=126$. Confirmed with sympy: $gg(x)=(x^2+5)^2+5$, and $gg(x)=126\Rightarrow(x^2+5)^2=121\Rightarrow x^2+5=11$ (the $-11$ root is rejected since $x^2+5\geqslant5>0$ always) $\Rightarrow x^2=6\Rightarrow x=\pm\sqrt6$ — note the mark scheme's own typeset root sign does not survive pdftotext extraction here (the same known artefact WMA13-verified-facts.md already documents for a different WMA13 question's nested root), so the final line was cross-checked independently with sympy rather than trusted from the raw text. The real, mark-scheme-named trap is distinct from every composed-argument trap already in the lesson: the mark scheme states plainly that the first method mark requires "an attempt to 'undo' $g$ once" toward finding $g(x)$, and separately — verbatim — "do not award this mark if the candidate believes that to find $x$ you only need to undo $g$ once. So, answers like $x=11$ ... will score 0,0,0 unless further work is seen." $x=11$ is exactly what solving $g(x)=126$ alone (one undoing, not two) produces — a plausible-looking wrong answer reachable by a genuine one-step misunderstanding of what "$gg$" asks for, not an arithmetic slip.

Micro-example, fresh numbers, self-verified. $p(x)=2x^2-3,\,x\geqslant0$; solve $pp(x)=51$. $pp(x)=2(2x^2-3)^2-3$; setting this to $51$ gives $(2x^2-3)^2=27$, so $2x^2-3=\sqrt{27}=3\sqrt3$ (rejecting $-3\sqrt3$ since $2x^2-3\geqslant-3$ but the equation still needs the positive root to give a real $x^2$ here — confirmed by checking both roots directly with sympy, only $+3\sqrt3$ yields $x^2\geqslant0$), so $x^2=\frac{3+3\sqrt3}{2}$, $x=\sqrt{\frac{3+3\sqrt3}{2}}\approx2.0244$. The one-undoing wrong answer would be solving $p(x)=51$ directly: $2x^2-3=51\Rightarrow x=\sqrt{27}=3\sqrt3\approx5.196$ — a different, plausible-looking number, exactly the shape of trap the real mark scheme names.

(b) A self-composed Möbius function stays Möbius, and its range is read from two limits, not calculus. Real anchor, January 2022 Q6(c), verified against the primary mark scheme (WMA13_01_2201_MS): $f(x)=\frac{5x-3}{x-4},\,x>4$; the question asks candidates to show $ff(x)$ collapses to another linear-fractional expression, $ff(x)=\frac{22x-3}{x+13}$ (confirmed exactly with sympy), then "deduce the range of $ff$" — worth 2 of the question's marks, and the real examiner report calls it "the most demanding part of the question," noting "many realised that 22 was significant... [but] very few stated 5." Independently confirmed with sympy: as $x\to4^+$ (the excluded domain boundary), $ff(x)\to5$; as $x\to\infty$, $ff(x)\to22$; since $f$ (and hence $ff$) is continuous and monotonic on $x>4$, the range is exactly $5<ff(x)<22$ — the two bounds come from the domain's own boundary behaviour, not from setting a derivative to zero. This is a genuinely different range-finding mechanism from every other range question in this document or the existing lesson, all of which read the range off a vertex or a monotonic half-line's starting value; here it comes from a two-sided limit at a domain that is open at both relevant ends.

#3.5 Composition at an algebraic argument

The lesson already carries the case where one side of an equation composes a number and the other composes an algebraic expression that must be collapsed via a stated sign constraint (Summer 2026 Q2(d), fci-mcq-6) — not reproduced here. A second, distinct real case exists where both sides carry a different algebraic argument into a different function, so the trap is not collapsing an argument but pairing each argument with the right function in the first place. Real anchor, June 2022 Q2(c), verified against the primary mark scheme (WMA13_01_2206_MS): $f(x)=\frac{5-x}{3x+2}$, $g(x)=2x-7$; solve $f\!\left(\frac1a\right)=g(a+3)$. The real examiner report is explicit about the failure mode this produces: "those who failed to gain any marks in this part typically set $f\left(\frac1a\right)=g\left(\frac1a\right)$ or $f(a)=g(a)$, or used the reciprocal for $f$" — i.e. copying one side's argument onto the other side, or (a second, independently named error) reading $f^{-1}$-style notation habits into a plain function evaluation and taking $\frac{1}{f(a)}$ instead of $f\!\left(\frac1a\right)$. Solved correctly, this collapses to $4a^2-a-2=0$, giving $a=\frac{1\pm\sqrt{33}}{8}$ — confirmed with sympy.

#3.6 The inverse-as-shortcut technique for solving $gf(a)=k$

Real anchor, January 2024 Q4(d), whose own mark scheme prints two equally credited routes side by side ("Way 1" and "Way 2") — worth stating explicitly because it is a real, mark-scheme-sanctioned choice of method, not just one correct path among several a student happened to find. With $f(x)=\frac{2x}{3x-5}$ and $g(x)=3+2\ln x$ from earlier parts of the same question, solving $gf(a)=5$: Way 1 substitutes $f(a)$ directly into $g$ and solves the resulting log equation in $a$ from scratch. Way 2 instead uses $g(f(a))=5 \iff f(a)=g^{-1}(5)$ — evaluates $g^{-1}(5)$ first (a single number, from the $g^{-1}$ already found in part (c)), then solves the simpler equation $f(a)=$ that number. Both are worth full marks on the real scheme and both reach the same $a=\frac{5e}{3e-2}$ (confirmed with sympy), but Way 2 only exists as an option because $g^{-1}$ had already been found earlier in the same question — when a question's structure hands you an inverse for one of the two composed functions, substituting into the inverse of the outer function and solving for the inner function's output can be a shorter route than composing forward and solving in one large step.

#3.7 Graphical: sketching $y=f^{-1}(x)$ from a sketch of $y=f(x)$

Not present anywhere in the existing lesson, whose own inverse content is entirely algebraic. Spec 1.2 explicitly names "inverse functions and their graphs" as its own examinable content, and one real question tests exactly the graphical skill with nothing algebraic attached. Real anchor, June 2023 Q4(b), verified against the primary mark scheme (WMA13_01_2306_MS): given a sketch of $y=f(x)$ for $f(x)=2x^2-5,\,x\geqslant0$ (an increasing curve starting at $(0,-5)$), sketch $y=f^{-1}(x)$ on the same axes. The real mark scheme's criteria are entirely about shape and position, not algebra: M1 for "a curve starting on the negative $x$-axis and passing through the positive $y$-axis, in quadrants 1 and 2 only" (i.e. the reflection of the given curve's start-point and general quadrant, not a plotted formula); A1 for "correct shape (curvature) and position... must be increasing (not bending back on itself) with decreasing gradient." The real examiner report names the two most common real errors directly: drawing "the reflection of the given graph in the y-axis to give a complete quadratic curve" (reflecting in the wrong axis entirely) and reflecting in the $x$-axis. Successful candidates, per the report, "appreciated that inverse functions are reflective in the line $y=x$" — precisely the mechanism the existing lesson already derives algebraically, here being marked purely as a drawing.

#3.8 Where $y=f(x)$ meets $y=f^{-1}(x)$: the $y=x$ shortcut versus the direct route

The single richest real exam angle located in this whole pass, because the same question's own mark scheme and examiner report between them document a genuine two-route fork with a stated, verified reason one route is structurally safer. Real anchor, June 2023 Q4(c), immediately following §3.7's sketch, on the same $f(x)=2x^2-5,\,x\geqslant0$: "the curve with equation $y=f(x)$ meets the curve with equation $y=f^{-1}(x)$ at the point $P$... find the exact $x$ coordinate of $P$." Because $f$ and $f^{-1}$ are reflections of each other in $y=x$, any point where they intersect off that line would have to come with a mirrored partner point also on both curves — for a curve that, like this one, is strictly monotonic, the only points that can be their own mirror image under reflection in $y=x$ are points already sitting on $y=x$, so solving $f(x)=x$ finds the same intersection point $f(x)=f^{-1}(x)$ would, without ever constructing $f^{-1}$ at all. Solving $f(x)=x$: $2x^2-5=x \Rightarrow 2x^2-x-5=0 \Rightarrow x=\frac{1+\sqrt{41}}{4}$ (rejecting the negative root, outside $f$'s domain $x\geqslant0$) — confirmed with sympy and matching the real mark scheme exactly.

The real mark scheme prices in the alternative directly: attempting the literal route, $f(x)=f^{-1}(x)$, "will lead a quartic ($8x^4-40x^2-x+45=0$ if correct) but will likely not lead to exact answers," and explicitly allows the independent set-up mark for reaching that quartic while denying the method and accuracy marks that need an exact solution from it. The real examiner report confirms this is exactly what happened at scale, not a hypothetical risk: "the students that were successful here realised that the point of intersection was on the line $y=x$... The majority however, attempted to find $f^{-1}(x)$, usually correctly, then put $f(x) = f^{-1}(x)$ and ended up with a quartic... were not able to find an exact answer to their quartic equation without significant extra work, only very rarely seen, to factorise it, so could not access the method nor the accuracy marks." Checked independently with sympy.factor: the quartic is reducible over the rationals — $8x^4-40x^2-x+45=(2x^2-x-5)(4x^2+2x-9)$ — and its first factor is exactly the $y=x$ shortcut's own quadratic (the second factor is the extraneous quadratic squaring introduces, from the branch $f(x)=-f^{-1}(x)$ rather than $f(x)=f^{-1}(x)$). An exact answer was never mathematically out of reach from the quartic; what the real evidence shows is that spotting a non-obvious factorisation of an unfamiliar quartic from a blank page, under exam time pressure, with no rational root to start from, is exactly the "significant extra work, only very rarely seen" the examiner report names — which is a genuinely different, more precise claim than "the quartic has no exact solution," and the one the primary sources actually support. The practical lesson is the same either way: the $y=x$ shortcut reaches the identical correct quadratic directly, without first constructing and then having to re-factorise the harder quartic that contains it.

#4. Full synthesis: every real exam angle located, and the trap taxonomy

Confirmed present in 9 of the 11 distinct WMA13 series this course's research has opened a primary document for (Oct 2020, Oct 2021, Jan 2022, Jun 2022, Oct 2022, Jun 2023, Jan 2024, Jan 2025, Oct 2024 — the original eight-series review plus the Oct 2024 gap-mining pass — together with Jan 2023 and Summer 2026, each opened separately as a named citation anchor): confirmed tested in Jan 2022, Jun 2022, Oct 2022, Jun 2023, Jan 2024, Oct 2024, Jan 2023 and Summer 2026 (all re-verified against primary sources across this document and the existing lesson combined), plus Oct 2021 (confirmed only via the already-verified quote in WMA13-verified-facts.md, since the archived PDF for that series could not itself be re-opened this pass — see the provenance note above). Not confirmed present in Oct 2020 or Jan 2025, on the evidence this course's research has gathered — absence of evidence in two series checked, not evidence that the topic never appeared in them.

Every trap below names its real source; none is invented to round out the list.

Six of these fourteen rows are genuinely new findings from the sources opened for this pass, not restatements of the existing lesson's six traps.

#What this document deliberately leaves to the existing files

The composite-domain and reflection-in-$y=x$ derivations, the diagram, both full worked chains, the marked solution, and the existing six trap-taxonomy entries with their own MCQs all live in functions-composition-and-inverses.ts and are not reproduced here. The compressed M/A/B skeletons for the Jan 2023 and Summer 2026 anchors live in codex-formula-and-method-bank.md. The bare frequency count (9 series, 6–9 marks) lives in signal-frequency-and-boundaries.md. This document's job was the self-inverse condition, the survey of real algebraic families beyond the restricted quadratic, the two self-composition mechanisms, the graphical and intersection-shortcut angles, and an honestly-sourced full trap taxonomy across all nine confirmed series.