# Functions

> IB Mathematics: Analysis and Approaches HL · IB AA HL 2021+
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u2-overview/
> Weight: 22-24% of overall exam

This unit covers all core function concepts from foundational definitions to properties of common function types, algebraic manipulation, and solving function-based problems. Functions are the foundation of all analysis and calculus in IB AA HL, making this one of the most critical core units.

**Prerequisites:** [Completion of IB AA HL Unit 1: Algebra](https://www.owlsprep.com/study/ib-math-aa-hl-u1-overview/)

## Learning objectives

- Define, classify, and analyze key properties of all common function types
- Manipulate functions algebraically, including composition, inverses, and transformations
- Solve equations and inequalities involving all core function families
- Apply function concepts to model real-world and abstract mathematical problems

## Unit at a glance

We progress sequentially from foundational definitions to advanced HL-only content, building your toolkit step by step. First we cover core definitions and basic manipulations, then we explore each major function family, before ending with general properties and problem solving.

Concepts from this unit are tested heavily across both Paper 1 and Paper 2, and are required for nearly all other units in the course, including calculus, differential equations, and modeling. Mastery of this unit is essential for achieving a high overall score.

Below are all sub-topics in this unit, ordered for sequential learning:
- [Function definitions and notation](https://www.owlsprep.com/study/ib-math-aa-hl-u2-function-definitions-and-notation/) — Learn what formally defines a function and standard notation for working with functions.
- [Domain and range](https://www.owlsprep.com/study/ib-math-aa-hl-u2-domain-and-range/) — Calculate the domain and range for any function, including functions with restricted domains.
- [Composite and inverse functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-composite-and-inverse-functions/) — Combine functions into composites and find the inverse of a function when it exists.
- [Transformations of functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-transformations-of-functions/) — Apply shifts, stretches, and reflections to any function and its graph.
- [Linear and quadratic functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-linear-and-quadratic-functions/) — Review and explore key properties of linear and quadratic functions.
- [Polynomial and rational functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-polynomial-and-rational-functions/) — Analyze roots, asymptotes, and graphs of polynomial and rational functions.
- [Exponential and logarithmic functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-exponential-and-logarithmic-functions/) — Explore the inverse relationship between exponential and logarithmic functions and their properties.
- [Basic trigonometric functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-basic-trigonometric-functions/) — Analyze graphs, domains, ranges, and key properties of the six basic trigonometric functions.
- [Inverse trigonometric functions (HL only)](https://www.owlsprep.com/study/ib-math-aa-hl-u2-inverse-trigonometric-functions/) — HL-only content covering definitions, restricted domains, and properties of inverse trigonometric functions.
- [Trigonometric identities](https://www.owlsprep.com/study/ib-math-aa-hl-u2-trigonometric-identities/) — Learn and apply core trigonometric identities to simplify expressions and solve equations.
- [Function properties: parity and periodicity](https://www.owlsprep.com/study/ib-math-aa-hl-u2-function-properties-parity-and-periodicity/) — Analyze general function properties of even/odd parity and periodicity.
- [Solving function equations and inequalities](https://www.owlsprep.com/study/ib-math-aa-hl-u2-solving-function-equations-and-inequalities/) — Solve equations and inequalities involving all function types covered in this unit.

## Common pitfalls

- **Wrong:** Forgetting to check the domain of composite or inverse functions
  - Why it fails: Most students only calculate the expression for the new function and ignore restricted domains, leading to incorrect full answers
  - Correct: Always state the full domain for any new function you create via composition or inversion
- **Wrong:** Mixing up the order of horizontal transformations for $f(kx + c)$
  - Why it fails: Order of operations for input transformations is often reversed, leading to incorrect shifts or stretches
  - Correct: Factor out the coefficient of $x$ to get $f(k(x + c/k))$, apply stretch/compression first, then shift
- **Wrong:** Ignoring domain constraints when solving logarithmic or rational equations
  - Why it fails: This leads to extraneous solutions that do not satisfy the original equation
  - Correct: Always check all solutions against the domain of the original function before finalizing your answer

## Cheatsheet

| Concept / Property | Key Rule / Formula |
| --- | --- |
| Inverse function | $f(f^{-1}(x)) = x$, graph is reflection of $f(x)$ over $y=x$ |
| General function transformation | $af(b(x-h)) + k$: $a$ = vertical stretch, $b$ = horizontal stretch, $h$ = horizontal shift, $k$ = vertical shift |
| Composite function | $(f \circ g)(x) = f(g(x))$, domain is $x \in dom(g)$ where $g(x) \in dom(f)$ |
| Logarithm core rules | $\log(ab) = \log a + \log b$, $\log(a/b) = \log a - \log b$, $\log a^b = b\log a$ |
| Pythagorean identity | $\sin^2 \theta + \cos^2 \theta = 1$ |
| Exponential-log conversion | $a^x = b \iff x = \log_a b = \frac{\ln b}{\ln a}$ |
| Even / odd functions | Even: $f(-x) = f(x)$ (symmetric over y-axis); Odd: $f(-x) = -f(x)$ (symmetric over origin) |
| Remainder Theorem | Remainder of $p(x)$ divided by $(x-a)$ equals $p(a)$ |

## What's next

Begin your learning of this unit with the first sub-topic on function definitions and notation, to build a strong foundational understanding. Once you complete all sub-topics in this Functions unit, you can progress to the first sub-topic of the next unit: Geometry and Trigonometry.

- [Function definitions and notation](https://www.owlsprep.com/study/ib-math-aa-hl-u2-function-definitions-and-notation/)
- [Domain and Range](https://www.owlsprep.com/study/ib-math-aa-hl-u2-domain-and-range/)
- [Composite and inverse functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-composite-and-inverse-functions/)

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