Lemmara logoLemmaraWorked solution

0606/12 Oct/Nov 2025, Question 4: Functions: One-One, Inverses and Restricted Domains

Cambridge IGCSE Additional Mathematics 0606 · Paper 12 (Non-calculator) · Non-calculator · 10 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Complete the square on 2x² + 5x + 3, locate the stationary point, then restrict the domain so the inverse exists and sketch both graphs with axis intercepts.

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

Part (a) · 2 marks

Rewrite 2x² + 5x + 3 as 2(x + a)² + b and find the constants a and b.

  1. 12x² + 5x + 3 = 2(x² + (5/2)x) + 3
  2. 2= 2[(x + 5/4)² − 25/16] + 3
  3. 3= 2(x + 5/4)² − 25/8 + 3
  4. 4= 2(x + 5/4)² − 1/8

Answer2(x + 5/4)² − 1/8 (a = 5/4, b = −1/8)

Part (b) · 2 marks

Use your completed-square form to state the coordinates of the stationary point of y = 2x² + 5x + 3.

  1. 1Minimum of 2(x + 5/4)² − 1/8 is when x + 5/4 = 0
  2. 2x = −5/4, y = −1/8

Answer(−5/4, −1/8)

Part (ci) · 1 mark

Given f(x) = 2x² + 5x + 3 for x ≥ p and that f⁻¹ exists, state the smallest possible p.

  1. 1f must be one-one, so its domain starts at the vertex
  2. 2p = −5/4

Answerp = −5/4

Part (cii) · 5 marks

With that p, draw y = f(x) and y = f⁻¹(x) on the blank axes, label each curve, and give where each meets the axes.

  1. 1f(x) = (2x + 3)(x + 1) = 0 ⇒ x = −1 (x = −3/2 is outside the domain)
  2. 2f(0) = 3, so f meets the y-axis at 3
  3. 3f starts at the vertex (−5/4, −1/8) and increases
  4. 4f⁻¹ is the reflection of f in y = x
  5. 5f⁻¹ meets the x-axis at 3 and the y-axis at −1

Answery = f(x): right half of the parabola from its vertex (−5/4, −1/8), crossing the x-axis at x = −1 and the y-axis at y = 3, increasing; y = f⁻¹(x): its reflection in y = x, starting at (−1/8, −5/4), crossing the x-axis at x = 3 and the y-axis at y = −1; each graph labelled

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: 0606/12/O/N/25 · 0606_w25_qp_12 · 0606 w25 12 · question 4.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.