Functions
SL2.1–SL2.7 (Functions)
This topic is really one idea repeated four times: something goes in, exactly one thing comes out, and you can describe the rule algebraically or as a graph. Every 'model this real-world situation' question — bacteria, vans, tanks, tides — is testing whether you can spot which of linear, quadratic, exponential or trig is the right shape, set up two equations, and solve. The rest (domain/range, composites, inverses, transformations) is vocabulary for describing graphs precisely enough that an examiner can mark it.
Overview
What counts as a function here
A function assigns exactly one output to every input in its domain — written or . Graphically, this is the vertical line test: any vertical line through the domain must cross the graph exactly once. Everything you do in this chapter — building a model, describing a graph's behaviour, stretching it, or restricting it to make an inverse exist — sits on top of that single rule.
- Domain = valid inputs; range = actual outputs — these are read off a sketch, not guessed from the formula alone.
- Four families dominate the SL syllabus: linear, quadratic, exponential and trigonometric (sin/cos/tan) — recognising which one fits a data pattern is half the battle in modelling questions.
- Your GDC is assumed throughout: graphing, finding intersections, and solving numerically are all fair game unless a question says 'show that' or 'algebraically'.
The shape of the chapter
Command terms that decide how you must answer
| Command term | What it demands | AO | Mark-earning move |
|---|---|---|---|
| Show that | derive the given result using clear algebra — a calculator-verified answer alone scores zero | AO2 | exact values (ln, surds) must appear in the working, not rounded decimals |
| Hence | use the result from the previous part directly | AO2 | re-deriving from scratch instead of using the given result loses marks even if the final answer is right |
| State | give the answer with no justification required | AO1 | one mark awarded for the correct value alone |
| Sketch | draw a graph with key features (intercepts, asymptotes, turning points) clearly shown | AO1/AO2 | marks are awarded per labelled feature, not for a technically accurate curve |
| Determine | find the value through systematic working, often via GDC | AO2 | quoting only a GDC output with no method shown can lose the method mark in multi-step questions |
Key point
Overview