Algebra and Functions
Core ideas
You must be fluent with polynomial and rational expressions, partial fractions, the factor and remainder theorems, inequalities, modulus equations and functions, and the behaviour of graphs (domain, range, asymptotes, transformations). Comfort with function notation and composition/inverse relationships is assumed.
Traps
- Dividing an inequality by a negative reverses the sign; squaring both sides can introduce spurious solutions, so always check against the original.
- Modulus equations split into cases; |x| < a means -a < x < a, but |x| > a means x > a or x < -a.
- When finding an inverse, state the domain restriction that makes the function one-to-one.
- Partial fractions need the correct form: a repeated factor (x-1)^2 requires both A/(x-1) and B/(x-1)^2.
- Vertical asymptotes occur where the denominator vanishes but the numerator does not.
Rule of thumb
For a rational inequality, move everything to one side, factorise the numerator and denominator, then test the sign in each interval between critical points (where numerator or denominator is zero). Never multiply through by an expression whose sign is unknown.
Worked example
Solve (x-1)/(x+2) ≥ 2. Rearrange: (x-1)/(x+2) - 2 ≥ 0, giving (x-1-2x-4)/(x+2) = (-x-5)/(x+2) ≥ 0, i.e. (x+5)/(x+2) ≤ 0. Critical points x = -5 and x = -2. Testing intervals: the expression is negative between -5 and -2, so the solution is -5 ≤ x < -2 (x = -2 excluded because the denominator is zero).