# Functions

> CIE IGCSE Additional Mathematics · CIE IGCSE Add Maths (0606)
> Source: https://www.owlsprep.com/study/cie-0606-u1-overview/
> Weight: Foundational topic; appears regularly on both Paper 1 and Paper 2

This unit builds core foundational knowledge of functions required for all higher-level topics in CIE IGCSE Additional Mathematics, covering basic definitions, domain/range, composites, inverses, and modulus graphs.

**Prerequisites:** Basic algebraic manipulation and substitution skills

## Learning objectives

- Define functions, distinguish them from relations, and calculate valid domain and range for common function types
- Perform composite function operations and identify conditions for inverse functions to exist
- Sketch, analyse, and solve problems involving modulus function graphs
- Apply function rules to answer standard IGCSE exam-style questions for this topic

## Unit at a glance

You will begin with foundational function concepts, learning to distinguish valid functions from relations, and calculate valid input (domain) and output (range) sets for linear, quadratic, rational, and root functions. You will then progress to more advanced function operations, including building composite functions, calculating inverses, and interpreting modulus function graphs as required for IGCSE exam questions.

Work through the following subtopics in sequence to build full mastery of this unit:
- [Functions, Domain and Range](https://www.owlsprep.com/study/cie-0606-u1-functions-domain-and-range/) — Learn to define functions, verify them against relation rules, and calculate domain and range for standard function types.
- [Composite, Inverse Functions and Modulus Graphs](https://www.owlsprep.com/study/cie-0606-u1-composite-inverse-functions-and-modulus/) — Master composite function operations, inverse function calculation, and sketching/analysing modulus function graphs for exam problems.

## Common pitfalls

- **Wrong:** Classifying one-to-many relations (e.g. $x=y^2$) as valid functions
  - Why it fails: Functions require every input value to map to exactly one unique output value, so one-to-many relations do not meet the function definition
  - Correct: Use the vertical line test for graphs, or verify every input maps to only one output for mapping diagrams and algebraic expressions
- **Wrong:** Calculating inverse functions for non one-to-one functions without domain restriction
  - Why it fails: Inverse functions only exist for one-to-one functions where every output maps to exactly one input value
  - Correct: Restrict the domain of non one-to-one functions (e.g. quadratics) to a region of strictly increasing or decreasing values before calculating the inverse
- **Wrong:** Including negative values in the range of a standard modulus function $|f(x)|$
  - Why it fails: The modulus operation converts all negative outputs to positive values, so $|f(x)|$ is always non-negative
  - Correct: Exclude all values less than 0 from the range of $|f(x)|$, unless a negative sign is placed outside the modulus operation

## Cheatsheet

| Concept/Formula | Subtopic Reference | Key Exam Note |
| --- | --- | --- |
| Function definition: Each input maps to exactly one output | Domain and Range | Use the vertical line test to validate functions from graphs |
| Domain = set of all valid input ($x$) values | Domain and Range | Exclude values that make denominators zero or square root arguments negative |
| Range = set of all valid output ($y$) values | Domain and Range | Use quick graph sketches to verify range calculations for complex functions |
| Composite function: $fg(x) = f(g(x))$ | Composite & Inverse Functions | Apply the inner function $g$ first, then substitute the result into the outer function $f$ |
| Inverse function $f^{-1}(x)$ existence condition: $f$ is one-to-one | Composite & Inverse Functions | Restrict domains of non one-to-one functions to find valid inverses if required |
| Modulus rule: $\|a\| = a$ if $a ≥ 0$, $\|a\| = -a$ if $a < 0$ | Modulus Graphs | Modulus graphs reflect all negative segments of $f(x)$ above the $x$-axis |

## What's next

Start with the first subtopic to build core function fundamentals, covering basic definitions, domain, and range. Once you have mastered these foundational concepts, move on to advanced function operations including composites, inverses, and modulus graphs to complete this unit. After finishing all subtopics here, you will be prepared to progress to the next unit on quadratic functions.

- [Quadratic Functions Unit Overview](https://www.owlsprep.com/study/cie-0606-u2-overview/)

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