# Functions

> IB Mathematics Applications and Interpretation HL · IB AI HL
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u2-overview/
> Weight: 20-25% of overall exam

This unit covers core definitions, properties, algebraic manipulation, graphing and real-world modelling of all key function types for IB AI HL, and is foundational to almost all other topics on the syllabus.

**Prerequisites:** [Basic algebra, indices, and coordinate geometry](https://www.owlsprep.com/study/ib-math-ai-hl-u1-overview/)

## Learning objectives

- Identify and classify all core function types tested in IB AI HL
- Calculate key properties (domain, range, roots, symmetry) for any function
- Manipulate functions algebraically including composition, inverse and transformations
- Solve function-based equations and inequalities using both algebraic and graphical methods
- Select and use appropriate functions to model real-world phenomena

## Unit at a Glance

We start with the core definition of a function and key properties that apply to all function types, before building up to core algebraic manipulations and graph transformations. We then explore each major family of functions (quadratic, exponential, sinusoidal) in detail, before ending with solving equations and applying functions to real-world modelling tasks. This sequence builds from foundational rules to applied skills, matching how questions are structured on IB exams.

Below are all sub-topics you will cover in this unit:
- [Concept of functions: domain, range, graphs](https://www.owlsprep.com/study/ib-math-ai-hl-u2-concept-of-functions-domain-range/) — Learn the formal definition of a function, how to calculate domain and range, and interpret function graphs.
- [Composition of functions and inverse functions](https://www.owlsprep.com/study/ib-math-ai-hl-u2-composition-of-functions-and-inverse/) — Understand how to combine multiple functions and reverse operations to find inverse functions.
- [Transformations of graphs](https://www.owlsprep.com/study/ib-math-ai-hl-u2-transformations-of-graphs/) — Learn how shifts, stretches and reflections transform the graph of any function.
- [Quadratic functions: roots, vertices, inequalities](https://www.owlsprep.com/study/ib-math-ai-hl-u2-quadratic-functions-roots-vertices-inequalities/) — Explore key properties of quadratics, find roots and vertices, and solve quadratic inequalities.
- [Exponential and logarithmic functions](https://www.owlsprep.com/study/ib-math-ai-hl-u2-exponential-and-logarithmic-functions/) — Explore the inverse relationship between exponential and logarithmic functions and their properties.
- [Sinusoidal functions and applications](https://www.owlsprep.com/study/ib-math-ai-hl-u2-sinusoidal-functions-and-applications/) — Learn how sinusoidal functions model periodic phenomena and solve real-world application problems.
- [Solving equations: algebraic and graphical methods](https://www.owlsprep.com/study/ib-math-ai-hl-u2-solving-equations-algebraic-and-graphical/) — Compare and use algebraic and graphical approaches to solve a wide range of function equations.
- [Odd and even functions, symmetry](https://www.owlsprep.com/study/ib-math-ai-hl-u2-odd-and-even-functions-symmetry/) — Identify symmetry properties of odd and even functions and use them to simplify function analysis.
- [Inverse functions: algebraic solution](https://www.owlsprep.com/study/ib-math-ai-hl-u2-inverse-functions-algebraic-solution/) — Practice advanced algebraic methods for finding inverse functions of common function types.
- [Modelling with functions](https://www.owlsprep.com/study/ib-math-ai-hl-u2-modelling-with-functions/) — Learn how to select, fit and validate functions to model real-world data and scenarios.

## Common pitfalls

- **Wrong:** Forgetting to restrict domain when finding inverse functions
  - Why it fails: Most functions are not one-to-one over their full domain, so inverses are only valid for restricted domains
  - Correct: Always confirm the original function is one-to-one over the given domain before calculating an inverse
- **Wrong:** Mixing up horizontal and vertical transformation rules
  - Why it fails: Horizontal transformations are counter-intuitive compared to vertical transformations, leading to common sign errors
  - Correct: Remember: $f(x - a)$ shifts right by $a$, while $f(x) + a$ shifts up by $a$

## Cheatsheet

| Concept | Key Formula / Rule |
| --- | --- |
| Function definition | A function maps each input in the domain to exactly one output in the range |
| Inverse function property | $f^{-1}(f(x)) = x$, $f(f^{-1}(x)) = x$, graph reflects over $y=x$ |
| Vertex of quadratic $f(x) = ax^2 + bx + c$ | $x = -\frac{b}{2a}$, $y = f(-\frac{b}{2a})$ |
| Exponent-log identity | $\log_a(a^x) = x$, $a^{\log_a x} = x$ |
| General sinusoidal function | $f(x) = a\sin(b(x - c)) + d$, amplitude $\|a\|$, period $\frac{2\pi}{\|b\|}$ |
| Odd/even function rule | Even: $f(-x) = f(x)$, symmetric over y-axis; Odd: $f(-x) = -f(x)$, symmetric over origin |

## What's next

Start your study of this unit with the first sub-topic covering the core concept of functions, domain and range. Once you complete all sub-topics in this unit, move on to the next unit on sequences and series, which builds directly on the core function concepts you will learn here.

- [Concept of functions: domain, range, graphs](https://www.owlsprep.com/study/ib-math-ai-hl-u2-concept-of-functions-domain-range/)
- [Sequences and Series](https://www.owlsprep.com/study/ib-math-ai-hl-u3-overview/)
- [Composition of functions and inverse functions](https://www.owlsprep.com/study/ib-math-ai-hl-u2-composition-of-functions-and-inverse/)

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From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-math-ai-hl-u2-overview/