Cambridge IGCSE · Additional Maths
Functions
A function is just a reliable machine: put a number in, get exactly one number out — and once you understand that one idea, inverses, domains, ranges, and composite functions all just become different questions about the same machine.
Summary — What This Chapter Covers
- Mappings vs functions — a function is a mapping where every input gives exactly one output (one-one or many-one only).
- Function notation — f(x), f: x ↦ ..., and how to evaluate f(3), f(a), f(x+3) by substitution.
- Domain — the set of allowed inputs. Range — the set of resulting outputs. The range depends entirely on the domain.
- The modulus function |x| — makes any input positive (the "absolute value").
- Inverse functions f⁻¹(x) — undoes what f did. Found by swapping x and y, then rearranging. Only exists if f is one-one.
- Graphs of inverses — always a reflection of y = f(x) in the line y = x.
- Composite functions fg(x) — apply g first, then f to the result. Order matters: fg(x) ≠ gf(x) generally.
- Domain/range of composites — g's range must fit inside f's domain, or fg(x) simply doesn't exist.
1. Language of Functions
1What is a Mapping? What is a Function?
Think of a mapping as an arrow diagram: it takes values from one set (the inputs) and connects them to values in another set (the outputs). That's it — any rule that connects inputs to outputs is a mapping.
A mapping can behave in different ways:
• One-to-one — every single input goes to its own unique output (nobody shares).
• Many-to-one — several different inputs are allowed to land on the output (like x² sending both 3 and −3 to 9).
• One-to-many and many-to-many — these exist too, but there's a problem with them...
The Key Idea
A
function is a mapping where every input produces
exactly one output — no exceptions. If just one input could give you two different outputs, it fails to be a function. That's why only
one-to-one and
many-to-one mappings count as functions — "one-to-many" breaks the rule immediately.
Why does this matter? Because a function needs to be predictable. If you tell a function "put in 5," it needs to give you back one definite answer every single time — not "well, it could be 12 or it could be −3." That reliability is the entire point of a function.
2Function Notation
Functions are usually named with letters like f, g, h. You'll see two main styles of notation, and they mean exactly the same thing:
How a function actually works: whatever sits inside the brackets replaces every x on the right-hand side. If you see f(3), every x becomes a 3. If you see f(a), every x becomes an a. If you see f(x + 3), every x becomes (x + 3) — brackets and all, which is the part that trips people up (see the example below).
Worked Example
Given f(x) = 3x² − 2x + 1, find (a) f(7), and (b) f(x + 3).
(a) Substitute x = 7 into every x:
f(7) = 3(7)² − 2(7) + 1 = 3(49) − 14 + 1 = 147 − 14 + 1
f(7) = 134
(b) Replace every x with the whole bracket (x + 3):
f(x+3) = 3(x+3)² − 2(x+3) + 1
Expand (x+3)² = x² + 6x + 9 first, then distribute the 3 and the −2:
= 3(x² + 6x + 9) − 2x − 6 + 1 = 3x² + 18x + 27 − 2x − 6 + 1
3