Lemmara logoLemmaraWorked solution

0606/11 May/Jun 2026, Question 6: Functions: One-One, Inverses and Restricted Domains

Cambridge IGCSE Additional Mathematics 0606 · Paper 11 (Non-calculator) · Non-calculator · 12 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

For f(x) = 3x² + 8x with x ≥ k and f⁻¹ existing, use the least possible k to find k, the range of f, f⁻¹, and work with g(x) = 3ˣ.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 2 marks

Find the smallest value of k for which f has an inverse.

  1. 1f must be one-one, so k is the x-coordinate of the vertex
  2. 2dy/dx = 6x + 8 = 0
  3. 3x = −4/3, so k = −4/3

Answerk = −4/3

Part (b) · 2 marks

State the range of f using the value of k found.

  1. 1f(−4/3) = 3(16/9) − 32/3 = −16/3
  2. 2Range: f(x) ≥ −16/3

Answerf ≥ −16/3

Part (c) · 3 marks

Derive an expression for f⁻¹(x).

  1. 1Let y = f⁻¹(x): x = 3y² + 8y
  2. 23y² + 8y − x = 0
  3. 3y = (−8 ± √(64 + 12x))/6
  4. 4y ≥ −4/3, so take +: f⁻¹(x) = (−8 + √(64 + 12x))/6

Answerf⁻¹(x) = (−8 + √(64 + 12x))/6

Part (di) · 1 mark

Write down the domain of the composite function gf.

  1. 1gf has the same domain as f, since the range of f (f ≥ −16/3) lies in the domain of g (x > −6)
  2. 2Domain: x ≥ −4/3

Answerx ≥ −4/3

Part (dii) · 4 marks

Solve gf(x) = 27 for x.

  1. 1gf(x) = 3^(3x² + 8x) = 27 = 3³
  2. 23x² + 8x = 3
  3. 33x² + 8x − 3 = 0, (3x − 1)(x + 3) = 0
  4. 4x = 1/3 or x = −3
  5. 5x = −3 is not in the domain x ≥ −4/3, so x = 1/3

Answerx = 1/3

Stuck on a step? Read the free lesson: Functions: One-One, Inverses and Restricted Domains

Practise more functions: one-one, inverses and restricted domains questions

Real 0606 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on functions: one-one, inverses and restricted domains

Also searched as: 0606/11/M/J/26 · 0606_s26_qp_11 · 0606 s26 11 · question 6.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.