What is the domain?
Which input values are allowed? Restrictions can come from denominators, roots, logarithms or the model itself.
MATHEMATICS · CORE LANGUAGE
Functions
A function assigns each allowed input exactly one output. Functions are the language of change, dependence and mathematical models — connecting algebra, geometry, calculus, probability and beyond.
THE FOUR QUESTIONS
Understand every function through the same lens.
What is the range?
Which output values can the function actually produce?
What does the graph do?
Intercepts, symmetry, growth, turning points and asymptotic behaviour reveal structure visually.
How can it be transformed?
Shift, scale, reflect and combine functions without losing track of what changes.
Can it be inverted?
Inverse functions reverse input and output when the original function is one-to-one on the chosen domain.
Can functions be composed?
Composition feeds the output of one function into another and builds more complex models.
FUNCTION LIBRARY
Learn the core building blocks.
f(x)
Function fundamentals & types
Definition, domain, codomain, range, notation and common families.
Start with functions →f(x)+aTransformations
Translations, stretches, compressions and reflections of graphs.
Explore transformations →f⁻¹Inverse & composition
Composition, invertibility and how to recover an input from an output.
Explore inverses →maxExtrema
Local and global maxima and minima, with links toward calculus.
Explore extrema →mx+bLinear functions & equations
See how straight-line graphs connect directly to linear equations.
Connect to linear equations →x²Quadratic functions & roots
Parabolas, zeros, discriminants and quadratic equations.
Connect to quadratics →WHY FUNCTIONS MATTER
Functions connect algebra to calculus.
Equations ask where expressions are equal. Functions describe entire relationships. Calculus then studies how those relationships change — through limits, derivatives and integrals.
Equations→Functions→Limits→Derivatives→Integrals