Functions & Inverses: The Complete Guide

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Functions carry the highest weighting (~35 marks) of any topic in Paper 1. Mastering them is non-negotiable.

The Two Big Ideas

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1. Transformation: Every function can be shifted, stretched, and reflected using parameters ($a$, $p$, $q$). Once you understand what each parameter does, you can sketch any function.
2. Inversion: Every function can be “reversed”. The inverse of $f$ is written $f^{-1}$ and is found by swapping $x$ and $y$. Graphically, an inverse is a reflection across the line $y = x$.

The Universal Form

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In Grade 12, every function follows a pattern:

Deep Dives (click into each)

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Each function below gets its own dedicated page where every parameter is explored, with worked examples and exam-style questions:

• The Logic of Inverses: The foundational concept behind all inverse functions.
• Linear Function: $y = mx + c$ — the building block of all functions.
• Quadratic Function (Parabola): $y = a(x-p)^2 + q$ — turning points, axis of symmetry, and domain restriction for inverses.
• Hyperbola: $y = \frac{a}{x-p} + q$ — asymptotes and the logic of division by zero.
• Exponential Function: $y = ab^{x-p} + q$ — growth, decay, and the link to finance.
• Logarithmic Function: $y = \log_b x$ — the inverse of exponentials and the laws of logs.
• Summary & Comparison Table: All functions side-by-side with their transformations, domains, ranges, and inverses.

Build on Your Lower-Grade Foundations

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If function work still feels shaky, revise these source lessons first:

• Grade 10 Functions — linear, quadratic, hyperbola, and exponential graph basics
• Grade 10 Fundamentals: Basic Algebra
• Grade 11 Functions — parameter shifts with $p$ and $q$
• Grade 11 Fundamentals: Function & Graph Basics
• Grade 11 Exponents & Surds — needed for exponential and logarithmic work

⏮️ Sequences & Series | 🏠 Back to Grade 12 | ⏭️ Finance, Growth & Decay