Functions

2026 Syllabus Objectives

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

  1. Understand functions, domain and range and use function notation
  2. Understand and find inverse functions f⁻¹(x)
  3. Form composite functions as defined by gf(x) = g(f(x))

What is a Function?

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A function is like a mathematical machine that takes numbers as inputs and transforms them into outputs using specific mathematical rules.

Think of it this way: imagine you have a blender (the function). You put in ingredients like milk, chocolate, and ice (the inputs), and the blender transforms them into a milkshake (the output). The function does the same thing with numbers.

Function Notation

When we write a function, we use special notation:

f(x) = ...

Here's what each part means:

  • f is the name of the function (we can also use other letters like g, h, or j)
  • x is the input value (the number going into the function)
  • f(x) is the output value (the result after the function processes x)

The equals sign is followed by the rule that tells you what to do with the input.

Examples of functions:

  • f(x) = 2x + 5 means "multiply the input by 2, then add 5"
  • g(x) = x² + 3 means "square the input, then add 3"
  • h(x) = 3x - 5 means "multiply the input by 3, then subtract 5"

Domain and Range

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Domain

The domain is the set of all possible input values (x-values) that you can put into a function.

Think of domain as "what numbers am I allowed to use?"

Important rules:

  • The domain always refers to x (not y or f(x))
  • Sometimes all numbers are allowed
  • Sometimes certain numbers must be excluded

Examples:

  1. f(x) = 2x + 5

    • Domain: all values of x
    • You can put any number into this function

  2. f(x) = 1/x

    • Domain: x ≠ 0 (x cannot equal zero)
    • Why? Because you cannot divide by zero

  3. f(x) = √x

    • Domain: x ≥ 0
    • Why? You cannot take the square root of a negative number

Range

The range is the set of all possible output values (y-values or f(x)-values) that come out of a function.

Think of range as "what numbers can the function produce?"

Important rules:

  • The range always refers to f(x) or y (not x)
  • The range depends on the domain

Examples:

  1. f(x) = x² with domain "all values of x"

    • Range: f(x) ≥ 0
    • Why? When you square any number (positive or negative), the result is always zero or positive

  2. f(x) = 3x + 2 with domain x > 0

    • When x = 0, f(0) = 2
    • Since all inputs are greater than 0, all outputs must be greater than 2
    • Range: f(x) > 2

Mapping Diagrams

A mapping diagram is a visual way to show how inputs are transformed into outputs by a function.

It consists of two columns:

  • Left column: Input values (domain)
  • Right column: Output values (range)
  • Arrows show which input maps to which output

Example: For f(x) = x + 3

Input          Output
  3      →       6
  4      →       7
  5      →       8
  x      →     x + 3

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