A-Level Maths: Algebra and Functions

Algebra and functions is the largest single section of the A-Level Mathematics prescribed content, and its techniques are assumed knowledge in every other topic — calculus, trigonometry and series all break down without fluent algebraic manipulation. The quiz, flashcards and printable written questions cover the main techniques of the section.

The manipulation questions start with algebraic fractions: simplifying (x² − 4)/(x² + 4x + 4) by factorising top and bottom, with the domain restriction that must accompany the cancellation. Indices and surds follow — evaluating (8x⁶)^(1/3) through fractional index laws, and combining √72 + √18 into a single surd, with the equivalent form √162 tested alongside 9√2 to check genuine understanding rather than one memorised route.

The factor theorem gets a full workout: identifying which of five candidate linear factors divide the cubic x³ − 2x² − 5x + 6 (each requiring an evaluation of f at the candidate root), and polynomial division of the perfect cube x³ − 3x² + 3x − 1 by (x − 1), where recognising the binomial structure shortcuts the working.

The functions questions cover the second-year (Year 13) material: composite functions with the order-of-application trap (fg means g first), inverse functions found by rearrangement, and the elegant self-inverse question — which functions satisfy f(f(x)) = x — where 1/x and 5 − x both qualify. The modulus function appears twice: solving |2x − 5| = 3 with its two cases, and reading properties (intercepts, symmetry) from f(x) = |x| − 2.

Each explanation shows the full manipulation, and the distractors encode the classic slips: cancelled terms that were never factors, composite functions applied in the wrong order, and sign errors in the two-case modulus split.

Simplify algebraic fractions by factorising, stating domain restrictions
Apply fractional and negative index laws and combine surds
Use the factor theorem and polynomial division on cubics
Form composite and inverse functions, recognising self-inverse cases
Solve modulus equations with the two-case method

Topic scope follows section B (Algebra and functions) of the DfE's prescribed AS and A level mathematics subject content — 100% common across all exam boards: indices, surds, quadratics, polynomial division and the factor theorem, composite and inverse functions, and the modulus function.

Sample question

Simplify the expression $\frac{x^2 - 4}{x^2 + 4x + 4}$ and state the domain restriction.

$\frac{x-2}{x+2}$, $x \neq -2$
$\frac{x-2}{x+2}$, $x \neq 2$
$\frac{x+2}{x-2}$, $x \neq -2$
$\frac{x-2}{x+2}$, $x \neq \pm 2$

See the answer

$\frac{x-2}{x+2}$, $x \neq -2$

Factorising the numerator gives $(x-2)(x+2)$ and the denominator gives $(x+2)^2$. Cancelling the common $(x+2)$ factor yields $\frac{x-2}{x+2}$. The domain must exclude values that make the original denominator zero, which is $x = -2$.

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