IB Mathematics: AA SL topic guide

Functions

Functions is a core part of IB Mathematics: AA SL. This guide connects the syllabus ideas behind Introduction to Functions, The Concept of a Function, Composite and Inverse Functions, Transformations of Graphs, Quadratic Functions and 2 more units, shows how they appear in worked problems, and points you to the formulas and full lessons needed for exam revision.

What you will learn

Functions syllabus outline

The units below follow the structure used in the full Study to Learn course. Use the outline to identify exactly which idea needs attention, then work through the public example before continuing to the complete lesson path.

2.0

Introduction to Functions

What You Will Learn in Functions

2.1

The Concept of a Function

Domain, Range & Function Notation · Key Features of Graphs

2.2

Composite and Inverse Functions

Composite Functions · Inverse Functions

2.3

Transformations of Graphs

Translations & Reflections · Stretches & Composite Transformations

2.4

Quadratic Functions

Forms of a Quadratic & the Discriminant · Applications of Quadratics

2.5

Rational, Exponential and Logarithmic Functions

The Reciprocal & Rational Functions · Exponential & Logarithmic Graphs

2.6

Solving Equations and Modelling with Functions

Solving Equations Graphically & Analytically · Modelling Real-World Data

Free worked preview

What You Will Learn in Functions

This complete preview comes from the Introduction to Functions unit. It introduces the core language, shows the method in context, and gives you a real example of the lesson quality before you create an account.

Introduction to Functions

Functions are the central organising idea of the AA course. They describe how one quantity depends on another — and nearly every topic that follows in calculus, statistics, and modelling is expressed in the language of functions. But a function is more than a formula: it is a mapping with a domain (what is allowed in), a range (what can come out), and a graph that tells a visual story about growth, decay, symmetry, and asymptotic behaviour. This section takes you from recognising basic parent functions through transforming them, composing them, inverting them, and ultimately using them to model the messy, non-linear relationships that appear in real data. The analytical, proof-oriented character of AA is especially visible here: you will prove the factor theorem, classify functions by their algebraic symmetry, and analyse asymptotic behaviour with the rigour that distinguishes HL from SL.
Section Map — What Is Covered
  • The Concept of a Function (2.1): domain and range, function notation, vertical line test, and reading intercepts, turning points and symmetry from a graph
  • Composite and Inverse Functions (2.2): chaining functions via (f∘g)(x)=f(g(x))(f\circ g)(x)=f(g(x)), finding inverses algebraically, and the reflection relationship in y=xy=x
  • Transformations of Graphs (2.3): translations f(x−h)+kf(x-h)+k, reflections, and stretches — the universal template for describing how any parent graph has been shifted and scaled
  • Quadratic Functions (2.4): general, factorised, and vertex forms; the discriminant Δ=b2−4ac\Delta=b^2-4ac; and optimisation problems solved via the vertex
  • Rational, Exponential and Logarithmic Functions (2.5): asymptotes of (ax+b)/(cx+d)(ax+b)/(cx+d), exponential growth and decay, and the logarithmic graph as an inverse exponential
  • Solving Equations and Modelling (2.6): analytical vs. graphical equation solving, and choosing the right function family to model real data by examining differences and ratios
  • Polynomial Functions (2.7, HL): the remainder theorem P(a)=remainderP(a)=\text{remainder} and factor theorem P(a)=0  ⟺  (x−a) is a factorP(a)=0 \iff (x-a) \text{ is a factor}, root multiplicity, and end behaviour of higher-degree polynomials
  • Further Rational and Special Functions (2.8, HL): oblique asymptotes when the numerator degree exceeds the denominator's by one, and the parity classification of odd and even functions
How to Approach This SectionFunctions are best understood by moving constantly between three representations: the algebraic formula, the graph, and a verbal description of what the function does. Do not get stuck in just one. When learning transformations, sketch each intermediate step — do not jump straight to the final equation. For rational functions, draw the asymptotes first as dashed lines, then locate intercepts, then sketch the curve approaching those asymptotes. The HL polynomial topics (factor theorem, oblique asymptotes) reward systematic working: test candidate roots by evaluating P(a) for factors of the constant term, and use polynomial division to uncover oblique asymptotes when the degree condition is met. If a problem asks find the values of k for which (x−k) is a factor\text{find the values of } k \text{ for which } (x-k) \text{ is a factor}, translate immediately to P(k)=0P(k)=0.

Use the formulas

Keep the formula booklet beside you

Revision becomes faster when you practise locating the right equation and understand what every variable means. The full reference is public and searchable on any device.

Open IB Math AA SL Formula Booklet →

Reviewed by the Study to Learn editorial team · Updated 2026-07-24