IBDP · Maths AI SL
Functions
SL2.1-2.5
28 min readStandard~25-30% of Paper 1 & 2 marks across the course; foundational for calculus and modelling questions
Functions is the topic examiners use to test whether you actually understand notation, not just whether you can push numbers through a calculator. Domain, range, inverse, composite — every one of these has a specific trap that costs marks year after year. Get the definitions rock solid first; the transformations and trig sections are just this same machinery dressed up in graphs.
Overview
A function is a rule that assigns exactly one output to each input in its domain. The whole SL Functions topic is built from three ideas repeated in different costumes: what values are you allowed to put in (domain), what values can come out (range), and what happens when you undo, combine, or reshape the rule.
- Domain — the set of -values a function accepts. Restricted by things you can't do: divide by zero, take the square root of a negative, take the log of zero or a negative number.
- Range — the set of -values the function actually produces, found from the graph's shape (min/max, asymptotes), not just guessed.
- Composite and inverse functions test whether you can chain notation correctly under exam pressure — this is where most marks are lost, not in the algebra itself.
- Transformations let you sketch or describe any new graph from a known parent graph without re-plotting from scratch.
- Trig functions bring periodicity into the mix — same domain/range logic, but now the range is bounded and repeats forever.
The shape of the chapter
Command terms that decide your method
| Command term | What it demands | AO | Mark-earning move |
|---|---|---|---|
| State | Give a result with no working required — usually 1 mark, don't waste time justifying it. | AO1 | Write the value/statement directly; extra working earns no more marks and costs time. |
| Show that | Prove a given result using algebra — the answer is given, so every line of working must be explicit. | AO2 | Marks for method lines, not just the final match to the given answer; skipping steps loses marks even if the answer matches. |
| Find | Produce a value or expression through calculation. | AO1/AO2 | Working must lead clearly to a final boxed/stated answer. |
| Determine | Find a result using given information, often needing you to select the correct method. | AO2 | State reasoning behind your method choice, especially for range/domain questions. |
| Hence | You must use the previous result/part — an independent correct method scores zero. | AO2 | Explicitly reference the earlier result in your working. |
| Justify | Give a reason supported by evidence (a calculation, a graph feature, a definition). | AO3 | A bare assertion with no reasoning scores 0 even if the conclusion is correct. |
Key point
exists only when is one-to-one on its stated domain. If a question restricts the domain (like for ), that restriction exists specifically to make the inverse exist — never ignore it.
Overview