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.
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 makes it smaller.
Practise Algebra, Functions & Proof — free games
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