Algebra (Higher) · Higher
Functions & Iteration
Function notation, composite and inverse functions, and iterative methods to solve equations.
Function notation f(x) is standard at Higher. You must find composites, inverses, and use iteration to approximate solutions.
Practice lesson for the WJEC Mathematics and Numeracy Double Award. This is not an official WJEC page.
Function notation
f(x) = 2x + 3 means “function f maps x to 2x + 3”.
f(5) = 2×5 + 3 = 13
**Composite** fg(x) means apply g first, then f.
**Inverse** f⁻¹ undoes f. For linear functions, rearrange y = f(x) to make x the subject, then swap.
Worked example
f(x) = 3x − 1. Find f⁻¹(x).
Solution
y = 3x − 1 → x = (y + 1)/3 **f⁻¹(x) = (x + 1)/3**
Iteration
Rearrange equation into x = g(x).
Choose a starting value x₀, then:
x₁ = g(x₀), x₂ = g(x₁), …
Continues until values settle (converge) or the question stops you.
Show enough decimal places as required.
Worked example
xₙ₊₁ = 1 + 1/xₙ, x₀ = 2. Find x₁ and x₂.
Solution
x₁ = 1 + 1/2 = 1.5 x₂ = 1 + 1/1.5 = 1 + 2/3 = **5/3 ≈ 1.667**
Practice questions
Try these without looking at the answers first.
Q1.f(x) = 2x + 5. f(3)?
Show answer
11
Q2.f(x) = 4x − 3. f⁻¹(x)?
Show answer
(x + 3)/4