Algebra and Functions
Algebra and Functions carries a large share of the Pancyprian Core Mathematics paper: quadratic and polynomial equations, inequalities, and the properties of functions and their graphs. The exam rewards precision - correct domains, correct signs on a number line, and a sketch whose features match the algebra you wrote.
Polynomials and equations
- Remainder theorem: the remainder when P(x) is divided by (x − a) is P(a); (x − a) is a factor exactly when P(a) = 0.
- For ax² + bx + c = 0, the discriminant is Δ = b² − 4ac and the roots are (−b ± √Δ)/(2a). Δ > 0 gives two distinct real roots, Δ = 0 a double root, Δ < 0 no real roots.
- Vieta: sum of roots = −b/a, product = c/a - often faster than solving.
Inequalities
- Move everything to one side, factorise, then build a sign table over the critical values.
- Never multiply an inequality by an expression whose sign is unknown; that is how denominator conditions disappear.
- Exclude every value that makes a denominator zero, and state the solution as an interval.
Functions
- Know domain, range, composition f∘g, and the inverse f⁻¹, which exists only when f is one-to-one and whose graph is the reflection of f in y = x.
- Transformations: f(x) + c shifts up, f(x + c) shifts left, af(x) stretches vertically, f(ax) compresses horizontally.
- aˣ and logₐx are inverse functions; both increase for a > 1 and decrease for 0 < a < 1.
Worked example
Solve (x − 1)/(x + 3) ≥ 2. Subtract 2 from both sides: (x − 1)/(x + 3) − 2 ≥ 0, so (x − 1 − 2x − 6)/(x + 3) ≥ 0, that is (−x − 7)/(x + 3) ≥ 0, equivalent to (x + 7)/(x + 3) ≤ 0. Critical values are −7 and −3, with x = −3 excluded because it zeroes the denominator. Testing the three intervals, the quotient is negative between them, so the solution is [−7, −3).
Traps that cost marks
- Cross-multiplying an inequality by (x + 3) without knowing whether it is positive.
- Including x = −3 in the final interval.
- Confusing f(2x) with 2f(x) when describing a transformation.
- Writing an inverse for a function that fails the one-to-one test on the stated domain.