Algebra and Functions
What this domain tests
Algebra and Functions underpins Paper 1: functions and graphs, composite and inverse functions, polynomials (remainder and factor theorems), partial fractions, binomial expansion of (1 + x)^n for rational n, and sequences and series including AP, GP and sums to infinity.
Core rules
- A function has an inverse only if it is one-one. Domain of f⁻¹ = range of f, and f(f⁻¹(x)) = x on that domain.
- Expand (1 + ax)^n with the general binomial term n(n-1)...(n-r+1)/r! (ax)^r; state validity |ax| < 1.
- A repeated linear factor in partial fractions needs a term for every power; divide first if the fraction is improper.
- A GP has a sum to infinity only when |r| < 1, with S∞ = a/(1 - r).
Worked example
Let f(x) = (3x - 2)/(x + 1), x ≠ -1. Find f⁻¹ and its domain. Set y = (3x - 2)/(x + 1). Then y(x + 1) = 3x - 2, so yx + y = 3x - 2 and x(y - 3) = -2 - y. Hence x = (y + 2)/(3 - y), giving f⁻¹(x) = (x + 2)/(3 - x). Since f(x) = 3 - 5/(x + 1), the range of f is all reals except 3, so the domain of f⁻¹ is x ≠ 3.
Traps
- Writing f⁻¹(x) as 1/f(x).
- Omitting the restricted domain of the inverse.
- Giving a binomial expansion without its validity interval.
- Sign slips when n is negative or fractional: in (1 - x)⁻¹ the coefficient of x² is +1, not -1.