Description

FUNCTIONS — Complete Study Guide (004)
Advanced Level Mathematics — Form Five

A comprehensive, exam-ready resource covering everything from the definition of a function to logarithmic graphs — all in one clearly organized document.

📘 What’s Inside
1. Introduction to Functions

Definition and key terminology: domain, range, independent/dependent variable, function notation

Visual representation (input → rule → output)

Function vs. Relation comparison table

2. Vertical Line Test

The test explained with clear diagrams (function vs. not a function)

3. Types of Functions

One-to-one (injective), Onto (surjective), and Bijective — each with diagrams and tests

Summary table linking each type to an example

4. Polynomial Functions

General form with all terms explained

5. Graphs of Linear Functions

Standard form, slope and y-intercept

Two fully worked graphing examples (positive and negative slope)

6. Graphs of Quadratic Functions

Shape analysis based on the value of *a*

Vertex formula and key features (axis of symmetry, vertex)

Three fully worked examples, including a quadratic with no x-intercepts

7. Graphs of Cubic Functions

Shape characteristics and key features

Three fully worked examples, including negative cubics

8. Graphs of Quartic Functions

W-shape vs. M-shape analysis

Two fully worked examples

9. Rational Functions

Definition and complete key-features table: domain, holes, vertical/horizontal/oblique asymptotes

10. Asymptotes — Complete Section

Vertical asymptotes (how to find, with visual)

Horizontal asymptotes — full rule table comparing degrees

Oblique asymptotes (when they occur, how to find by long division)

11. Worked Examples: Rational Functions

7 fully worked examples covering:

Finding holes and vertical asymptotes

Finding vertical and horizontal asymptotes

Finding oblique asymptotes (long division shown)

Graphing with vertical asymptotes

Graphing with oblique asymptotes

Graphing with a hole

Full rational function analysis with range restrictions

12. Composite Functions

Definition, notation, and visual representation (inner vs. outer function)

7 fully worked examples including:

Basic composition (both directions)

Composition with itself

Evaluating compositions (two methods)

Composition with square roots and logarithms

Solving composition equations

Composition of sets

Finding the inner function g(x) given f(g(x))

13. Graphs of Composite Functions

Fully worked example sketching both (f∘g)(x) and (g∘f)(x)

14. Exponential Functions

Definition, general form, and key properties table

Visual representations (increasing vs. decreasing)

Three fully worked graphing examples, including multiple exponential functions on the same axes

15. Logarithmic Functions

Definition as the inverse of exponential

Key properties table

Visual representations

Four fully worked examples, including transformed logarithms

16. Chapter Summary

Key concepts table

Summary of graph shapes (linear, quadratic, cubic, quartic, rational, exponential, logarithmic)

Key points to remember for every function type

17. Quick Reference Card

Vertical line test, polynomial, rational, composite, exponential, logarithmic, and all transformations on one card

18. Practice Questions

Five sections: Identifying Functions, Polynomials, Rational Functions, Composite Functions, and Exponential/Logarithmic

✅ Why This Guide Works
Every function type fully graphed — from linear to logarithmic, with plotted points and drawn curves

Asymptotes demystified — vertical, horizontal, and oblique with clear rules and long division shown

Composite functions from every angle — evaluation, equations, sets, and finding inner functions

Exam-focused — practice questions by topic, key points to remember, and a quick reference card

Complete coverage — from “what is a function?” to transformed logarithmic graphs in a single document

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