A function f is defined by f(x) = (x^2 - 2x + 1) / (x - 3), x ∈ ℝ, x ≠ 3. (a) Express f(x) in the form Ax + B + C / (x - 3), where A, B and C are integers to be found. (b) Hence, or otherwise, find the range of f.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. M1: For attempting algebraic long division or synthetic division, or equating coefficients after multiplying out. Must go as far as Ax+B + C/(x-3).
2. A1: For A = 1 and B = 1 correctly found.
3. A1: For C = 4 correctly found. (x + 1 + 4/(x - 3))
4. M1: For differentiating f(x) or the equivalent expression found in (a) and setting dy/dx = 0.
5. A1: Correct derivative, e.g., 1 - 4(x-3)^-2 or ((x-3)(2x-2) - (x^2-2x+1)(1)) / (x-3)^2.
6. M1: For solving dy/dx = 0 for x. Leads to (x-3)^2 = 4 so x-3 = ±2, giving x=5 or x=1.
7. A1: For correctly identifying the y-coordinates of the turning points: f(5) = 6 and f(1) = 2. Then stating the range as f(x) ≤ 2 or f(x) ≥ 6.