# Functions

> IB Mathematics: Applications and Interpretation SL · IB AI SL
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u2-overview/
> Weight: 22-24% of overall exam

This unit covers core function concepts, key function families, and applied modeling techniques that form the foundation of most quantitative problem solving in IB AI SL. You will connect algebraic rules to graphical behavior and use functions to solve real problems.

**Prerequisites:** Algebra fundamentals (Unit 1)

## Learning objectives

- Recognize the definition of a function and correctly determine the domain and range of common function types
- Perform core function operations including composition and calculation of inverse functions
- Graph and analyze key function families: linear, quadratic, exponential, and logarithmic
- Predict the effect of transformations on function graphs and equations
- Select and use appropriate functions to model real-world quantitative scenarios

## 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](https://www.owlsprep.com/study/ib-math-ai-sl-u2-concept-of-function-domain-range/) — Learn the formal definition of a function and how to find domain and range from equations and graphs.
- [Inverse functions](https://www.owlsprep.com/study/ib-math-ai-sl-u2-inverse-functions/) — Understand the relationship between a function and its inverse, and find inverses algebraically and graphically.
- [Linear functions and their graphs](https://www.owlsprep.com/study/ib-math-ai-sl-u2-linear-functions-and-their-graphs/) — Review and master properties of linear functions, their graphs, and core applications.
- [Quadratic functions, roots, and vertices](https://www.owlsprep.com/study/ib-math-ai-sl-u2-quadratic-functions-roots-and-vertices/) — Analyze quadratic functions, find roots, identify vertices, and interpret their graphs.
- [Exponential and logarithmic functions](https://www.owlsprep.com/study/ib-math-ai-sl-u2-exponential-and-logarithmic-functions/) — Explore the inverse relationship between exponential and logarithmic functions and their key properties.
- [Transformations of function graphs](https://www.owlsprep.com/study/ib-math-ai-sl-u2-transformations-of-function-graphs/) — Learn how translations, stretches, and reflections affect function graphs and their equations.
- [Real-world modeling with functions](https://www.owlsprep.com/study/ib-math-ai-sl-u2-real-world-modeling-with-functions/) — Apply your knowledge of functions to build and interpret models for real-world quantitative scenarios.

## Common pitfalls

- **Wrong:** Forgetting to restrict the domain of a composite function to the requirements of both inner and outer functions.
  - Why it fails: Algebraic simplification can hide invalid input values that are not allowed for the inner function.
  - Correct: Always cross-check that inputs are valid for both the inner and outer function when finding the domain of a composite.
- **Wrong:** Confusing horizontal transformation rules, e.g. assuming $f(x+2)$ shifts the graph right 2 units.
  - Why it fails: Changes inside the function (to the $x$ term) work opposite to intuition about direction.
  - Correct: Remember: changes inside $f()$ affect the x-axis in the opposite direction of the sign.
- **Wrong:** Treating all relations as functions without checking the definition first.
  - Why it fails: Not all graphs or relations satisfy the one-input-one-output requirement for functions, leading to incorrect inverse calculations.
  - Correct: Always verify a relation is a function with the vertical line test or formal definition before proceeding.

## Cheatsheet

| Concept / Key Formula | Description |
| --- | --- |
| Function definition | A relation where each input $x$ in the domain maps to exactly one output $y$ |
| Domain of $f(g(x))$ | All $x$ in domain of $g$ where $g(x)$ is in the domain of $f$ |
| Inverse function property | $f^{-1}(f(x))=x$; graph of inverse is reflection over $y=x$ |
| Linear function | $f(x) = mx + c$, $m$ = slope, $c$ = y-intercept |
| Quadratic vertex form | $f(x) = a(x-h)^2 + k$, vertex at $(h,k)$ |
| Exponential function | $f(x) = ab^x$, $a$ = initial value, $b$ = growth/decay factor |
| Logarithm identity | $\log_b(b^x) = x$ and $b^{\log_b(x)} = x$ |
| General transformation | $f(x-h) + k$ shifts $f(x)$ right $h$ units and up $k$ 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.

- [Concept of function, domain, range, and graph](https://www.owlsprep.com/study/ib-math-ai-sl-u2-concept-of-function-domain-range/)
- [Unit 3: Geometry and Trigonometry](https://www.owlsprep.com/study/ib-math-ai-sl-u3-overview/)

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