# Functions

> IB Mathematics AA SL · IB AA SL
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u2-overview/
> Weight: 20-23% of total exam score

This unit covers core function concepts, algebraic operations, transformations, and analysis of all key function types tested in IB AA SL. Functions are the foundation for mathematical modeling and appear across all exam sections.

**Prerequisites:** Basic algebraic manipulation and equation solving; Coordinate geometry fundamentals from Unit 1

## Learning objectives

- Identify and correctly represent functions using standard notation, domain, and range conventions
- Perform algebraic operations on composite and inverse functions, and relate their properties graphically
- Apply graph transformation rules to sketch and analyze functions of all types covered in the syllabus
- Analyze key properties of common function types to solve mathematical and real-world modeling problems

## Unit at a Glance

Functions are the backbone of quantitative modeling in mathematics, and this unit builds sequentially from core definitions to applied analysis of the most common function types you will use throughout the course. We start with foundational definitions and notation, then build to advanced algebraic operations and graphical transformations, before diving into the specific properties of each major function type that appear on IB exams.

Below are all sub-topics in this unit, ordered to build your knowledge progressively:
- [Function concepts, domain, range and notation](https://www.owlsprep.com/study/ib-math-aa-sl-u2-function-concepts-domain-range-and/) — Learn the formal definition of a function and how to calculate domain and range for any relation.
- [Composite and inverse functions](https://www.owlsprep.com/study/ib-math-aa-sl-u2-composite-and-inverse-functions/) — Combine functions into composites and find inverses both algebraically and graphically.
- [Transformations of graphs of functions](https://www.owlsprep.com/study/ib-math-aa-sl-u2-transformations-of-graphs-of-functions/) — Master shifts, stretches, reflections and combinations of transformations for any function graph.
- [Linear functions and their graphs](https://www.owlsprep.com/study/ib-math-aa-sl-u2-linear-functions-and-their-graphs/) — Review and extend properties of linear functions, gradients, and their graphical representations.
- [Quadratic functions, roots and discriminant](https://www.owlsprep.com/study/ib-math-aa-sl-u2-quadratic-functions-roots-and-discriminant/) — Analyze quadratic functions and use the discriminant to determine the number of real roots.
- [Reciprocal and rational functions](https://www.owlsprep.com/study/ib-math-aa-sl-u2-reciprocal-and-rational-functions/) — Explore properties of reciprocal and simple rational functions including asymptotes.
- [Exponential and logarithmic functions and graphs](https://www.owlsprep.com/study/ib-math-aa-sl-u2-exponential-and-logarithmic-functions-and/) — Learn key properties, graphs, and the inverse relationship between exponential and logarithmic functions.

## Common pitfalls

- **Wrong:** Forgetting to restrict the 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 and restrict the domain before calculating an inverse
- **Wrong:** Mixing up horizontal and vertical transformation rules
  - Why it fails: Horizontal transformations follow the opposite of intuitive direction, while vertical transformations follow intuition
  - Correct: Remember $f(x+a)$ shifts left by $a$ units, while $f(x)+a$ shifts up by $a$ units
- **Wrong:** Swapping the order of composition in $f(g(x))$
  - Why it fails: Function composition is not commutative, so order changes the final result
  - Correct: Always substitute the inner function ($g$) into the outer function ($f$) in the correct order

## Cheatsheet

| Concept / Formula | Key Description |
| --- | --- |
| $y=f(x)$ function notation | Relates an input $x$ to exactly one output $y$ |
| Composite function: $(f\circ g)(x) = f(g(x))$ | The output of $g$ becomes the input of $f$ |
| Inverse function property: $f^{-1}(f(x))=x$ | Graph of $f^{-1}(x)$ is reflection of $f(x)$ over $y=x$ |
| Quadratic discriminant: $\Delta = b^2-4ac$ | Determines number of real roots: $\Delta>0$ (2), $\Delta=0$ (1), $\Delta<0$ (0) |
| Transformation: $af(x+b)+c$ | Horizontal shift $-b$, vertical stretch $a$, vertical shift $c$ |
| Exponential-log inverse relationship: $\log_a(a^x)=x$, $a^{\log_a x}=x$ | Logarithms are the inverse of exponential functions |
| Rational function asymptotes | Vertical asymptotes where denominator = 0, horizontal asymptotes for end behavior |

## What's next

Begin with the first sub-topic below to build your foundational understanding of functions, which all subsequent topics in this unit rely on. Once you complete all sub-topics in this unit on functions, you will move on to the first sub-topic of the next unit on trigonometry.

- [Function concepts, domain, range and notation](https://www.owlsprep.com/study/ib-math-aa-sl-u2-function-concepts-domain-range-and/)
- [Composite and inverse functions](https://www.owlsprep.com/study/ib-math-aa-sl-u2-composite-and-inverse-functions/)
- [Transformations of graphs of functions](https://www.owlsprep.com/study/ib-math-aa-sl-u2-transformations-of-graphs-of-functions/)

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