IGCSE Function Notation: Composite, Inverse & Domain Guide

In higher-level mathematics, we move away from simple equations like y = 2x + 3 and adopt Function Notation, written as f(x) = 2x + 3. This notation is a powerful tool that allows us to describe complex relationships, combine multiple rules, and reverse processes.

This guide covers Topic 3.2 of the IGCSE Functions & Graphs Hub. We will demystify the three pillars of Higher Tier functions: Evaluation, Composition (combining functions), and Inversion (undoing functions).

1. Understanding Function Notation

Think of a function, f, as a machine. It takes an input (x), applies a rule, and produces an output.

• f(x): Read as "f of x". It represents the output when the input is x.
• Evaluate f(5): Replace every x in the formula with 5.

Domain and Range (H)

You must be able to identify valid inputs and possible outputs.

• Domain: The set of all possible inputs (x-values).
  Restricted Domain: You cannot divide by zero. For \(f(x) = \frac{1}{x-2}\), the domain is \(x \neq 2\).
• Range: The set of all possible outputs (y-values).
  Example: For \(f(x) = x^2\), the range is \(f(x) \geq 0\) (squares are always positive).

2. Composite Functions: fg(x) (H)

A composite function combines two functions. The notation fg(x) means "apply g first, then apply f to the result."

The Golden Rule: Inside-Out

Always work from the right to the left.
fg(x) means f( g(x) ).
1. Take the expression for g(x).
2. Put it INSIDE f(x).

3. Inverse Functions: f⁻¹(x) (H)

The inverse function reverses the operation of the original function. If f turns input A into output B, then f⁻¹ turns B back into A.

The "Swap and Solve" Method

1. Write the function as y = ...
2. Swap x and y (so y becomes x, and x becomes y).
3. Rearrange the formula to make the new y the subject.
4. Replace y with the notation f⁻¹(x).

Key Notation Summary

Step-by-Step Worked Examples

Real-World Application (Global Context)

Functions are essential in computer programming, particularly in data security and conversion.

Scenario: Currency Conversion & Fees

Imagine an international payment system.

• Function c(x) converts USD ($) to Euros (€): c(x) = 0.9x.
• Function f(x) applies a €5 service fee: f(x) = x - 5.

The final amount a customer receives depends on the order.
fc(x) means convert first, then subtract fee: 0.9x - 5.
cf(x) means subtract fee first, then convert: 0.9(x - 5) = 0.9x - 4.5.
Banks use composite functions to ensure fees are applied in the correct order. To reverse a transaction (refund), they must calculate the inverse function.

Exam Technique and Common Pitfalls

1. The Order of Composite Functions

This is the most common error.
• fg(x) means g first, then f.
• gf(x) means f first, then g.
Think of it like reading "f of g of x".

2. Inverse Notation Confusion

Do not confuse f⁻¹(x) with indices.
• In algebra, x⁻¹ means 1x.
• In functions, f⁻¹(x) means the inverse function, not the reciprocal.

3. Domain Errors

When finding domains of rational functions like f(x) = 1/(x-5), always set the denominator to zero to find the excluded value. Here, x ≠ 5.

Mastering advanced function algebra is essential for securing Grade 9 marks, particularly in the final questions of the paper.

Summary Checklist and Next Steps

Checklist:

• [ ] I can evaluate a function by substituting a number (e.g., f(3)).
• [ ] I can find a composite function fg(x) by substituting g into f.
• [ ] I can find an inverse function f⁻¹(x) by swapping variables and rearranging.
• [ ] (H) I can identify restricted values in a domain (e.g., dividing by zero).

Next Steps:
Now that you understand function notation, you are ready to visualize these relationships. Topic 3.3 Linear Graphs explores the visual representation of f(x) = mx + c.