Functions
AA HL Topic 2 — Functions (SL2.1–SL2.8; AHL2.9–AHL2.13). The Trigonometric Functions subtopic draws on Topic 3, Geometry & Trigonometry (SL3.7; AHL3.8–3.9), but is grouped here because exam papers test it with the same transformation toolkit as the rest of Functions.
Functions is the toolkit topic — everything else in the course eventually gets expressed as an you have to sketch, transform, invert or feed into a real-world story. At HL you bolt three heavier layers onto the SL toolkit: rational and polynomial functions with their asymptote/root machinery, odd/even and self-inverse structure, and a deeper reciprocal-and-inverse layer on the trig functions. Exam questions almost never ask 'find the domain' in isolation — they bury it inside bacteria, tides, satellite dishes or cooling coffee, and mark you on translating the context into algebra, extracting the right feature, then reading the number back into the story.
Overview — the shape of the topic
Every Functions question is one of four moves in disguise: build a model from a description, extract a property (domain, inverse, asymptote, root), transform a known graph, or handle a periodic (trig) scenario. HL adds algebraic depth — rational/polynomial structure, odd/even symmetry, reciprocal and inverse trig — but the exam habits are identical across SL and HL: identify the family, extract the feature with correct notation, then interpret it in the units of the question.
The shape of the chapter
Command terms that decide how you answer
| Command term | What it demands | AO | Mark-earning move |
|---|---|---|---|
| Show that | Derive the given result, showing every algebraic line | AO2 | No marks for quoting the answer — the working IS the mark scheme; skip a step and you lose it even with a correct final line. |
| Sketch | Graph with correct shape, asymptotes and intercepts labelled; scale not required | AO1/2 | Missing a labelled asymptote or intercept costs a mark even if the overall shape is right. |
| Draw | Accurate, to-scale graph, usually on provided grid/graph paper | AO1 | Unlike sketch, examiners check plotted points against a scale — wrong scale loses marks. |
| State | One-line answer, no justification expected | AO1 | 1 mark means 1 line — a full derivation here wastes time and earns nothing extra. |
| Hence | Must use the immediately preceding result or method | AO2 | A correct answer from a different method still loses marks if it ignores the 'hence'. |
| Hence or otherwise | Any valid method is accepted | AO2/3 | Free to use a fresh method if the previous result is awkward to apply. |
| Determine | Find a value/expression showing clear working | AO2 | A bare answer without supporting working can lose method marks even if correct. |
| Interpret | Explain the meaning of a value in the context given | AO3 | Must reference the scenario's units/story — restating the number in different words earns nothing. |
Key point
Overview