Unit Overview

Functions

CIE IGCSE Additional Mathematics· 5 min read 📊 Foundational topic; appears regularly on both Paper 1 and Paper 2

1. 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

Learn to define functions, verify them against relation rules, and calculate domain and range for standard function types.

★★⏱ 10 min

Composite, Inverse Functions and Modulus Graphs

Master composite function operations, inverse function calculation, and sketching/analysing modulus function graphs for exam problems.

★★★⏱ 12 min

2. Common Pitfalls

Wrong move:

Classifying one-to-many relations (e.g. $x = y^{2}$ ) as valid functions

Why:

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 move:

Use the vertical line test for graphs, or verify every input maps to only one output for mapping diagrams and algebraic expressions

Wrong move:

Calculating inverse functions for non one-to-one functions without domain restriction

Why:

Inverse functions only exist for one-to-one functions where every output maps to exactly one input value

Correct move:

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 move:

Including negative values in the range of a standard modulus function $∣ f (x) ∣$

Why:

The modulus operation converts all negative outputs to positive values, so $∣ f (x) ∣$ is always non-negative

Correct move:

Exclude all values less than 0 from the range of $∣ f (x) ∣$ , unless a negative sign is placed outside the modulus operation

3. Quick Reference 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: $f g (x) = f (g (x))$	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 $a \geq 0$ , $∣ a ∣ = - a$ 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