Ch 8 · 2.12 Functions · IGCSE Mathematics 0580
Topic 8 of 9 in Algebra and Graphs — IGCSE Mathematics 0580 — 18 lessons.
Introduction
A function is a rule that turns each input into exactly one output, like a machine: "multiply by 3, then subtract 5". This lesson covers function notation, how to run a function backwards (its inverse), and how to feed one function into another (a composite). (Syllabus section 2.12; in IGCSE 0580 it is section E2.13.)
Key ideas
f(x) = 3x - 5 means "the function f takes x to 3x - 5"; f(4) = 7. The domain is the set of inputs, the range the set of outputs. A mapping diagram shows arrows from inputs to outputs.
🧑🏫 Teacher's explanation
Think of a function as a specialized machine in a factory. You drop an input (the raw material) into the machine, it performs a specific task, and spits out an output (the finished product). The set of all possible raw materials you can use is called the domain, and the set of all possible finished products is the range.
📖 Definitions
🔍 Worked example — Easy
Given f(x) = 4x - 7, find the value of f(5).
🔍 Worked example — Standard
Given f(x) = 2x + 3 and g(x) = x^2, find the composite function gf(x) in its simplest form.
🔍 Worked example — Exam-style
Given f(x) = racx + 52, find the inverse function f^-1(x).
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
f(x) = 3/x + 2 and g(x) = 3x + 5. Find fg(x) as a single fraction: fg(x) = 3/(3x + 5) + 2 = 3/3x + 7.
🔍 Worked example — Easy
Given f(x) = 3/x + 2 and g(x) = 3x + 5, find fg(x) as a single fraction.
🔍 Worked example — Standard
Given h(x) = 2x^2 + 3 for x ≥ 0, find h^-1(x).
🔍 Worked example — Exam-style
Given f(x) = 3x - 5, find the value of x for which f(x) = f^-1(x).
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 11; f^-1(x) = 5 - x/2. (2) fg(x) = (x + 4)^2; gf(x) = x^2 + 4. (3) g^-1(x) = 2x - 1; g(3) = 2, g(2) = 1.5. (4) 1/2x + 1 = dfrac15 ⇒ x = 2.