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4MA1/1HR Jan 2019, Question 23: Inverse Functions

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1HR (Higher) · Calculator · 6 marks

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The question, in short

For f(x) = √(x² + k²)/x with x > 0 and k positive, determine a value p satisfying an inverse condition, and use g(x) = x² to express a in terms of k.

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

Part (a) · 3 marks

Work out the number p such that applying the inverse of f to p returns k.

  1. 1Let y = f(x) = √(x² + k²) / x
  2. 2Square both sides: y² = (x² + k²)/x², so x²y² = x² + k²
  3. 3Collect the x terms: x²(y² − 1) = k², so x = k/√(y² − 1)
  4. 4So f⁻¹(p) = k/√(p² − 1)
  5. 5f⁻¹(p) = k means k/√(p² − 1) = k, so √(p² − 1) = 1
  6. 6p² − 1 = 1, so p² = 2 and p = √2 as p > 0

Answer√2

Part (b) · 3 marks

When the composite gf evaluated at x = a equals k with k > 1, derive a formula for a involving k.

  1. 1g(x) = x², so gf(a) = (f(a))²
  2. 2(f(a))² = (a² + k²)/a², so (a² + k²)/a² = k
  3. 3a² + k² = ka², so k² = a²(k − 1)
  4. 4a² = k²/(k − 1), so a = k/√(k − 1) as a > 0

Answerk/√(k − 1)

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