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4MA1/2H Jan 2022, Question 25: Inverse Functions

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 2H (Higher) · Calculator · 5 marks

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The question, in short

Given g(x) = 5 + 6x − x² for x ≥ 3, find its inverse function and the domain of that inverse.

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

Part (a) · 4 marks

Find g⁻¹(x) in the form g⁻¹ : x ↦ …, using the restriction x ≥ 3.

  1. 1Let y = 5 + 6x − x²
  2. 2Rearrange: x² − 6x + (y − 5) = 0
  3. 3Use quadratic formula: x = [6 ± √(36 − 4(y − 5))]/2
  4. 4Simplify: x = [6 ± √(36 − 4y + 20)]/2
  5. 5x = [6 ± √(56 − 4y)]/2
  6. 6x = [6 ± 2√(14 − y)]/2
  7. 7x = 3 ± √(14 − y)
  8. 8Since domain of g is x ≥ 3, range of g⁻¹ must give values ≥ 3
  9. 9So take the positive root: x = 3 + √(14 − y)
  10. 10Interchange x and y: y = 3 + √(14 − x)
  11. 11g⁻¹ : x ↦ 3 + √(14 − x)

Answer3 + √(14 − x)

Part (b) · 1 mark

State the domain of g⁻¹.

  1. 1The expression under the square root must be non-negative: 14 − x ≥ 0
  2. 2So x ≤ 14

Answerx ≤ 14

Stuck on a step? Read the free lesson: Inverse Functions

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