1.3 Function Notation

2026 📋 Syllabus Objectives

By the end of this section, you should be able to:

  • Recognise and correctly use function notation in its different forms
  • Understand and apply the notation f(x), including examples such as f(x) = 2eˣ
  • Understand and apply the mapping notation f : x ↦ lg x
  • Recognise and use inverse function notation: f⁻¹(x)
  • Recognise and use composite function notation: fg(x), which means f(g(x))
  • Recognise and use repeated function notation: f²(x), which means f(f(x))

📝 Main Content

https://cambridgeassistant.com

What Is a Function?

Before we look at notation, let's quickly recall what a function is.

A function is a rule that takes an input value and produces exactly one output value. Think of it like a machine: you put a number in, and exactly one number comes out.

For example, the rule "multiply by 3 and then add 1" is a function. If you put in 2, you always get 7. If you put in 5, you always get 16. There is never any doubt about the output.

The Two Ways to Write a Function

In O Level Additional Mathematics, functions are written in two main ways. Both mean exactly the same thing — they are just different styles of notation.

Style 1: f(x) Notation

This is the most common way. It is written as:

f(x)=some rule involving xf(x) = \text{some rule involving } x
  • The letter f is the name of the function. (Other letters like g, h can also be used.)
  • The x inside the brackets is the input — the number you put in.
  • The right-hand side tells you what to do to the input.

Examples:

NotationWhat it means
f(x) = 2eˣThe function f takes x and gives 2 times eˣ as the output
g(x) = x² + 3The function g takes x, squares it, then adds 3
h(x) = 5x − 1The function h takes x, multiplies by 5, then subtracts 1

💡 Note on eˣ: The letter e is a special mathematical number (approximately 2.718). You will learn more about it in the topic on exponentials. For now, just treat eˣ as a rule applied to x.

💡 Note on lg x: The notation lg x means the logarithm of x to base 10. You will study logarithms in depth later. For now, recognise it as a valid rule for a function.

Style 2: Mapping Notation — f : x ↦ …

This style uses the arrow symbol ↦ (read as "maps to") and is written as:

f:x↦some rule involving xf : x \mapsto \text{some rule involving } x

Examples:

NotationHow to read itWhat it means
f : x ↦ lg x, for x > 0"f maps x to lg x, for x greater than 0"The function f takes any positive x and gives lg x as the output
f : x ↦ 2x + 3"f maps x to 2x + 3"The function f takes x, doubles it, then adds 3

⚠️ Important: The two styles are interchangeable. Writing f(x) = lg x and f : x ↦ lg x mean exactly the same thing. You must be able to read and write both forms.

⚠️ Domain restriction: When a domain restriction is given (for example, for x > 0), it is part of the function definition. It tells you which input values are allowed. For f : x ↦ lg x, the restriction x > 0 exists because you cannot take the logarithm of zero or a negative number.

Sign in to view full notes