Functions

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 $(f\circ g)(x)=f(g(x))$, finding inverses algebraically, and the reflection relationship in $y=x$
• Transformations of Graphs (2.3): translations $f(x-h)+k$, 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 $\Delta=b^2-4ac$; and optimisation problems solved via the vertex
• Rational, Exponential and Logarithmic Functions (2.5): asymptotes of $(ax+b)/(cx+d)$, 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 $P(a)=\text{remainder}$ and factor theorem $P(a)=0 \iff (x-a) \text{ is a factor}$, 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