Functions
AI HL Topic 2: Functions (SL2.1-SL2.5 function basics, modelling, key features; AHL2.6-AHL2.7 composite/inverse functions and transformations); sinusoidal graph work cross-refs AHL3.7 (Geometry & Trigonometry, circular functions)
Functions is the topic examiners lean on to test whether you actually understand algebra or just recognise patterns — every rational-function inverse, every transformed graph, every sinusoidal tank/tide model is really asking the same three questions: can you undo a mapping, can you track what , , , do to a graph, and can you read amplitude/period/asymptote straight off a picture. Get comfortable with those three and this topic stops being three subtopics and becomes one recurring skill.
Overview
At HL, Functions sits on top of the SL content (domain, range, key features, modelling) and adds two genuinely new tools: composite/inverse functions with domain restrictions, and a fully general transformation equation . Both get tested through rational functions (asymptotes, self-inverse shortcuts) and through trigonometric models (amplitude, period, principal axis).
- Functions and their properties: domain/range, one-to-one vs many-to-one, inverse functions, composite functions, asymptotic behaviour of rational functions.
- Transformations: the single equation covers every translation, stretch and reflection you'll ever be asked to apply.
- Trigonometric functions: sin/cos/tan graphs, amplitude-period-vertical shift, and solving trig equations over a restricted domain — the go-to tool for anything periodic.
The shape of the chapter
Command terms that decide your mark
| Command term | What it demands | AO | Mark-earning move |
|---|---|---|---|
| Write down | State a result with no working required — read directly from GDC or given info. | AO1 | 1 mark for the correct value/equation alone; there's no method mark to fall back on if it's wrong. |
| Show that | Prove a given result algebraically — the answer is already known, so every step must be visible. | AO2 | Marks awarded per legitimate step; jumping straight to the printed answer loses marks even if it's correct. |
| Hence | Use the previous part's result directly. | AO2 | A fresh method that ignores the earlier answer scores zero, even if mathematically valid. |
| Justify | Give an algebraic or logical reason, not a description of the graph. | AO2/AO3 | "It looks one-to-one on the graph" with no algebra typically earns 0 of the available marks. |
| Sketch | Show general shape and key features — intercepts, asymptotes, endpoints — not an accurate scale drawing. | AO1 | Marks for each correctly labelled feature; a missing asymptote or endpoint loses a mark even if the curve shape is right. |
| Determine / Find | Obtain the answer via calculation or GDC, with working shown. | AO1/AO2 | Method marks are available even with a wrong final value, provided the working is legible. |
Key point
Overview