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4MA1/1H Jan 2020, Question 21: Composite Functions

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1H (Higher) · Calculator · 5 marks

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The question, in short

Given f(x) = x² − 2x and g(x) = x + 3, with h(x) = fg(x) for x ≥ −2, find an expression for the inverse function h⁻¹(x).

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

Solution · 5 marks

  1. 1f(g(x)) = f(x + 3) = (x + 3)² − 2(x + 3)
  2. 2(x + 3)² − 2(x + 3) = x² + 6x + 9 − 2x − 6
  3. 3x² + 4x + 3
  4. 4Let y = h(x) = x² + 4x + 3
  5. 5Complete the square: y = (x + 2)² − 4 + 3
  6. 6y = (x + 2)² − 1
  7. 7(x + 2)² = y + 1
  8. 8x + 2 = ±sqrt(y + 1)
  9. 9Since x ≥ −2, take the positive root: x + 2 = sqrt(y + 1)
  10. 10x = −2 + sqrt(y + 1)
  11. 11h^(−1)(x) = −2 + sqrt(x + 1)

Answer−2 + √(x + 1)

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