Find the values of a and b so that f(x) = {3, if x ≤ 1; ax + b, if 1 < x < 4; 9, if x ≥ 4} is a continuous function.

1. 1 · Write what is given and what is to be found. Given: f in three pieces, with break points at x = 1 and x = 4. To find: a and b.
2. 2 · Write the rule. f is continuous at c when LHL = RHL = f(c). Away from 1 and 4, each piece is a polynomial, so it is continuous.
3. 3 · Continuity at x = 1.
   • f(1) = 3 and LHL = 3
   • RHL = lim (x → 1⁺) (ax + b) = a + b
   • So a + b = 3 ... (1)
4. 4 · Continuity at x = 4.
   • f(4) = 9 and RHL = 9
   • LHL = lim (x → 4⁻) (ax + b) = 4a + b
   • So 4a + b = 9 ... (2)
5. 5 · Solve the equations. (2) − (1): 3a = 6, so a = 2. From (1), b = 1.
6. 6 · Check and box the answer. At x = 1: 2 + 1 = 3. At x = 4: 8 + 1 = 9. Answer: a = 2, b = 1.

Scoring tip: Name both break points and write LHL, RHL and f(c) at each. Examiners give a mark for each point.