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4MA1/2H Jan 2023, Question 25: Inverse Functions

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

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The question, in short

Given f(x) = 3x² − 12x + 7 for x ≤ 2, find the inverse function f⁻¹(x) in the form f⁻¹(x) = …

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

Solution · 4 marks

  1. 1Let y = 3x² − 12x + 7
  2. 2Complete the square: y = 3(x² − 4x) + 7
  3. 3y = 3((x − 2)² − 4) + 7
  4. 4y = 3(x − 2)² − 12 + 7
  5. 5y = 3(x − 2)² − 5
  6. 6Swap x and y: x = 3(y − 2)² − 5
  7. 7Solve for y: x + 5 = 3(y − 2)²
  8. 8(x + 5)/3 = (y − 2)²
  9. 9y − 2 = ±√((x + 5)/3)
  10. 10Since x ≤ 2, y − 2 ≤ 0, so take negative root: y − 2 = −√((x + 5)/3)
  11. 11y = 2 − √((x + 5)/3)
  12. 12f⁻¹(x) = 2 − √((x + 5)/3)

Answer2 − √((x + 5)/3)

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