Algebra and Functions
Core concepts
A candidate must be fluent with polynomials, the factor and remainder theorems, partial fractions (including repeated and improper cases), inequalities, modulus equations and inequalities, and the behaviour of composite and inverse functions. Know that a function has an inverse only when it is one-to-one on its stated domain, and be ready to restrict a domain to make it so.
Traps
- Squaring both sides of a modulus or surd equation introduces extraneous roots: always check against the original.
- When solving a rational inequality, multiplying by an unknown-signed expression loses the direction of the inequality. Rearrange to compare with zero and use a sign table or sketch instead.
- For partial fractions with a repeated linear factor, remember every power appears in the decomposition.
- Domain of a composite function is not just the inner domain; the output of the inner function must lie in the domain of the outer one.
Worked example
Solve |2x − 5| < 7. This means −7 < 2x − 5 < 7, so −2 < 2x < 12, giving −1 < x < 6. Check the endpoints: at x = −1 the modulus is 7, excluded; at x = 6 it is 7, excluded. Rule of thumb: for |A| < k write the double inequality; for |A| > k split into two separate inequalities joined by 'or'.