Relations, Functions and Algebra (Foundational Overlap)
Relations, Functions and Algebra is the foundation layer the rest of the paper silently assumes. It is rarely labelled as a separate question, but lost marks in later chapters usually trace back here: set notation, intervals and inequalities, modulus, logarithms, and the algebra of quadratic equations.
Essentials
- Sets: A ∪ B, A ∩ B, A − B and complement; n(A ∪ B) = n(A) + n(B) − n(A ∩ B); De Morgan's laws.
- Intervals: (a, b) is open, [a, b] closed. A function is defined only where its expression is real - denominators nonzero, even roots non-negative, logarithms positive.
- Modulus: |x| = a means x = ±a only when a ≥ 0; |x| < a is equivalent to −a < x < a; |x| > a gives x < −a or x > a.
- Logarithms: log(xy) = log x + log y, log(xⁿ) = n log x, log_b a = log a/log b, with x > 0.
- Quadratic ax² + bx + c = 0: discriminant D = b² − 4ac. D > 0 gives distinct real roots, D = 0 equal real roots, D < 0 complex conjugate roots. Sum of roots = −b/a, product = c/a.
Worked example
Solve |2x − 3| ≤ 5. This is equivalent to −5 ≤ 2x − 3 ≤ 5. Adding 3 throughout gives −2 ≤ 2x ≤ 8. Dividing by the positive 2 gives −1 ≤ x ≤ 4, so the solution set is [−1, 4].
Traps
- Multiplying or dividing an inequality by a negative number without flipping the sign.
- Squaring both sides of an inequality without checking signs.
- Taking a logarithm of a negative or zero quantity.
- Writing x = ±a for |x| = a when a is negative; there is no solution.