A-Level Maths: Algebra, Functions & Proof Revision

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What does x^(1/2) mean?

Algebra, Functions & Proof is a core part of A-Level Maths. Revise the key concepts and common mistakes below, then lock them in with the free games.

Key concepts

FunctionA mapping f: X → Y assigning exactly one y to each x in the domain X. Written y = f(x).

Domain & rangeDomain is the set of valid inputs; range is the set of outputs produced. For f(x)=√x, domain x≥0, range f(x)≥0.

Composite functionfg(x) means apply g first, then f: fg(x) = f(g(x)). Order matters; fg(x) ≠ gf(x) in general.

Inverse functionf⁻¹ reverses f: f(f⁻¹(x)) = x. Exists only if f is one-to-one. Domain of f⁻¹ = range of f.

Modulus function|x| = x if x≥0, else −x. Graph is V-shaped, vertex at origin. |f(x)|=k gives two equations: f(x)=±k.

Transformationsy = f(x)+a translates up by a; y = f(x+a) translates left by a; y = af(x) stretches vertically; y = f(ax) compresses horizontally.

Proof by deductionBuild a chain of logical steps from known facts to a conclusion.

Proof by contradictionAssume the opposite, derive a contradiction, so the original must be true.

Counter-exampleA single example that disproves a universal claim.

Even and oddEven = 2k, odd = 2k+1 for integer k. Useful in number proofs.

Rational vs irrationalRational = p/q with p,q integers, q≠0; otherwise irrational.

Factor theoremIf P(a)=0 then (x-a) is a factor of polynomial P(x).

Quadratic formax²+bx+c=0 with a≠0 is the standard quadratic.

DiscriminantΔ = b²-4ac tells you the nature of roots.

What your exam board covers

Where Algebra, Functions & Proof sits in each exam board's specification. Tap a board to see what it lists.

AQAAlgebra and functionsSpec 7357

Part of: Pure mathematics

Source: AQA A-level Mathematics (7357) specification — Subject content, accessed 23 June 2026.

AQAProofSpec 7357

Part of: Pure mathematics

Source: AQA A-level Mathematics (7357) specification — Subject content, accessed 23 June 2026.

CCEAAlgebra and functionsSpec 2210

Part of: AS 1 — Pure mathematics

Source: CCEA A-Level Mathematics specification, accessed 23 June 2026.

EdexcelAlgebra and functionsSpec 9MA0

Part of: Pure mathematics

Source: Pearson Edexcel A level Mathematics (9MA0) specification, accessed 23 June 2026.

EdexcelProofSpec 9MA0

Part of: Pure mathematics

Source: Pearson Edexcel A level Mathematics (9MA0) specification, accessed 23 June 2026.

OCRAlgebra and functionsSpec H240

Part of: Pure mathematics

Source: OCR A Level Mathematics A (H240) specification, accessed 23 June 2026.

OCRProofSpec H240

Part of: Pure mathematics

Source: OCR A Level Mathematics A (H240) specification, accessed 23 June 2026.

Learn Algebra, Functions & Proof step by step

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Common mistakes to avoid

Questions where students often pick the tempting wrong answer — make sure you know the right one:

What does x^(1/2) mean?✗ x^(1/2) means x divided by 2 — half of x. ✓ The square root of x (the positive root when x is positive).

If the discriminant of a quadratic is negative, what does it tell you about real roots?✗ There is one repeated root — the curve just touches the x-axis. ✓ There are no real roots — the graph does not cross the x-axis.

Nature of roots if Δ<0✗ Two real roots ✓ No real roots

Does multiplying always make a number bigger?✗ Yes, multiplication always increases a number. ✓ No — multiplying by a number between 0 and 1, or by a negative number, can make it smaller (e.g. 6 × 0.5 = 3 and 6 × (−2) = −12); one counter-example disproves the claim.

Practise Algebra, Functions & Proof — free games

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▶ Two Truths, One Lie ▶ Reveal Race ▶ Connections Grid ▶ Falling Words ▶ Word Web

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Algebra, Functions & Proof

Key concepts

What your exam board covers

Common mistakes to avoid

Practise Algebra, Functions & Proof — free games

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