Unit Overview
Functions
IB Mathematics: Applications and Interpretation SLΒ· 5 min read π 22-24% of overall exam
1. Unit at a Glance
We start with the fundamental definition of a function, building up to core operations like composition and inverses before moving into detailed study of the most common function families you will use throughout this course. This unit balances procedural skills (such as finding domains, graphing functions, and solving for roots) with applied skills that align with AI SL's focus on practical problem solving: building models from real data.
The topic sequence builds incrementally: you will master core concepts first, then explore specific function types, learn transformations that apply to all functions, and end by applying all your knowledge to real-world modeling. Each topic relies on concepts from earlier in the unit, so it is best to work through them in order.
This unit includes the following sub-topics:
Concept of function, domain, range, and graph
Learn the formal definition of a function and how to find domain and range from equations and graphs.
β β β± 6 min
Composite functions
Learn how to form, evaluate, and find the domain of composite functions.
β β β β± 5 min
Inverse functions
Understand the relationship between a function and its inverse, and find inverses algebraically and graphically.
β β β β± 5 min
Linear functions and their graphs
Review and master properties of linear functions, their graphs, and core applications.
β β± 4 min
Quadratic functions, roots, and vertices
Analyze quadratic functions, find roots, identify vertices, and interpret their graphs.
β β β± 7 min
Exponential and logarithmic functions
Explore the inverse relationship between exponential and logarithmic functions and their key properties.
β β β β± 8 min
Transformations of function graphs
Learn how translations, stretches, and reflections affect function graphs and their equations.
β β β β± 6 min
Real-world modeling with functions
Apply your knowledge of functions to build and interpret models for real-world quantitative scenarios.
β β β β β± 9 min
2. Common Pitfalls
Wrong move:
Forgetting to restrict the domain of a composite function to the requirements of both inner and outer functions.
Why:
Algebraic simplification can hide invalid input values that are not allowed for the inner function.
Correct move:
Always cross-check that inputs are valid for both the inner and outer function when finding the domain of a composite.
Wrong move:
Confusing horizontal transformation rules, e.g. assuming shifts the graph right 2 units.
Why:
Changes inside the function (to the term) work opposite to intuition about direction.
Correct move:
Remember: changes inside affect the x-axis in the opposite direction of the sign.
Wrong move:
Treating all relations as functions without checking the definition first.
Why:
Not all graphs or relations satisfy the one-input-one-output requirement for functions, leading to incorrect inverse calculations.
Correct move:
Always verify a relation is a function with the vertical line test or formal definition before proceeding.
3. Quick Reference Cheatsheet
Concept / Key Formula | Description |
|---|---|
Function definition | A relation where each input in the domain maps to exactly one output |
Domain of | All in domain of where is in the domain of |
Inverse function property | ; graph of inverse is reflection over |
Linear function | , = slope, = y-intercept |
Quadratic vertex form | , vertex at |
Exponential function | , = initial value, = growth/decay factor |
Logarithm identity | and |
General transformation | shifts right units and up units |
What's Next
This unit builds step-by-step, so we recommend starting with the first sub-topic on core function definitions, domain, and range. Work through each sub-topic in order to build your skills gradually. After completing all sub-topics in this unit on functions, you will progress to the next unit covering geometry and trigonometry.