IGCSE Composite and Inverse of Functions: Complete Guide for Cambridge IGCSE Mathematics

IGCSE composite and inverse functions are advanced function topics in Cambridge IGCSE Mathematics that appear in Paper 4 (Extended). Mastering function composition, finding inverse functions, and function operations is essential for achieving top grades in IGCSE exams.

This comprehensive IGCSE composite and inverse functions guide covers everything you need to know, including composite functions (fg(x)), inverse functions (f⁻¹(x)), worked examples, common exam questions, and expert tips from Tutopiya’s IGCSE maths tutors. We’ll also show you how to avoid the most common mistakes that cost students valuable marks.

🎯 What you’ll learn: By the end of this guide, you’ll know how to find composite functions, find inverse functions, and apply these skills to solve problems in IGCSE exams.

Already studying with Tutopiya? Practice these skills with our dedicated IGCSE Additional Mathematics practice deck featuring exam-style questions and instant feedback.

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Why IGCSE Composite and Inverse Functions Matter

IGCSE composite and inverse functions are essential advanced function topics. Here’s why they’re so important:

• High frequency topic: Composite and inverse function questions appear regularly in IGCSE Extended papers
• Foundation skill: Essential for advanced mathematics and function analysis
• Exam weight: Typically worth 6-10 marks per paper
• Real-world applications: Used in function modeling and transformations
• Problem-solving skills: Develops understanding of function relationships

Key insight from examiners: Students often confuse composite and inverse functions or make errors with function notation. This guide will help you master these systematically.

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Understanding Composite Functions

Composite function fg(x) means apply g first, then apply f to the result.

fg(x) = f(g(x))

Example 1: If f(x) = 2x + 1 and g(x) = x², find fg(x)

Solution: fg(x) = f(g(x)) = f(x²) = 2(x²) + 1 = 2x² + 1

Answer: 2x² + 1

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Understanding Inverse Functions

Inverse function f⁻¹(x) undoes what f(x) does.

If f(a) = b, then f⁻¹(b) = a

Example 2: If f(x) = 2x + 3, find f⁻¹(x)

Solution: Let y = 2x + 3 Swap x and y: x = 2y + 3 Solve for y: x - 3 = 2y, so y = (x - 3) / 2 Therefore: f⁻¹(x) = (x - 3) / 2

Answer: f⁻¹(x) = (x - 3) / 2

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Verifying Inverse Functions

f(f⁻¹(x)) = x and f⁻¹(f(x)) = x

Example 3: Verify that f⁻¹(x) = (x - 3) / 2 is the inverse of f(x) = 2x + 3

Solution: f(f⁻¹(x)) = f((x - 3) / 2) = 2((x - 3) / 2) + 3 = x - 3 + 3 = x ✓

Answer: Verified

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Common Examiner Traps

• Order confusion - fg(x) means apply g first, then f
• Inverse method errors - Swap x and y, then solve for y
• Notation errors - f⁻¹(x) is the inverse, not 1/f(x)

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Practice Questions

Question 1

If f(x) = x + 2 and g(x) = 3x, find fg(x)

Solution: fg(x) = f(g(x)) = f(3x) = 3x + 2

Answer: 3x + 2

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Frequently Asked Questions About IGCSE Composite and Inverse Functions

What is a composite function?

A composite function fg(x) means apply g first, then apply f to the result: fg(x) = f(g(x)).

What is an inverse function?

An inverse function f⁻¹(x) undoes what f(x) does. If f(a) = b, then f⁻¹(b) = a.

How do I find the inverse of a function?

Swap x and y, then solve for y. The result is f⁻¹(x).

How do I verify an inverse function?

Check that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

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Related IGCSE Maths Resources

Strengthen your IGCSE Mathematics preparation with these comprehensive guides:

• IGCSE Graphs of Functions: Complete Guide - Master function graphs
• IGCSE Domain and Range: Complete Guide - Master domain and range
• IGCSE Maths Revision Notes, Syllabus and Preparation Tips - Complete syllabus overview, topic breakdown, and revision strategies
• IGCSE Past Papers Guide - Access free IGCSE past papers and exam resources

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Next Steps: Master IGCSE Composite and Inverse Functions with Tutopiya

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