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4MA1/1H Jan 2023, Question 18: Composite Functions

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

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The question, in short

For f(x) = k/x with integer k, find its inverse, evaluate g at −2, and form the composite hg in simplest form.

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

Part (a) · 1 mark

Write f⁻¹(x) in the requested form for f(x) = k/x, x ≠ 0.

  1. 1f(x) = k/x
  2. 2Let y = k/x
  3. 3Multiply both sides by x: xy = k
  4. 4Divide both sides by y: x = k/y
  5. 5Swap x and y: y = k/x
  6. 6f⁻¹(x) = k/x

Answerk/x

Part (bi) · 1 mark

Calculate the value of g(−2) for g(x) = 2 − 3x⁴.

  1. 1g(x) = 2 − 3x⁴
  2. 2g(−2) = 2 − 3(−2)⁴
  3. 3(−2)⁴ = 16
  4. 43 × 16 = 48
  5. 52 − 48 = −46

Answer−46

Part (bii) · 2 marks

Express hg(x) in simplest form using h(x) = 3x/(2 − x) and g(x) = 2 − 3x⁴.

  1. 1g(x) = 2 − 3x⁴ and h(x) = 3x/(2 − x)
  2. 2hg(x) = h(2 − 3x⁴)
  3. 3hg(x) = 3(2 − 3x⁴) ÷ (2 − (2 − 3x⁴))
  4. 4Numerator: 3(2 − 3x⁴) = 6 − 9x⁴
  5. 5Denominator: 2 − 2 + 3x⁴ = 3x⁴
  6. 6hg(x) = (6 − 9x⁴)/(3x⁴)
  7. 7Divide the numerator and the denominator by 3
  8. 8hg(x) = (2 − 3x⁴)/x⁴

Answer(2 − 3x⁴)/x⁴

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