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0580/41 May/Jun 2024, Question 9: Composite Functions

Cambridge IGCSE Mathematics 0580 · Paper 41 (Extended) · Calculator · 12 marks

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The question, in short

Given f(x) = 4x + 1, g(x) = 6 − 2x and h(x) = 3^(x − 2), evaluate values, find an inverse, and solve equations involving these functions.

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

Part (ai) · 1 mark

Calculate f(3).

  1. 1f(3) = 4 × 3 + 1 = 13

Answer13

Part (aii) · 1 mark

Calculate gf(3), applying g to the value of f(3).

  1. 1gf(3) = g(13) = 6 − 2 × 13 = −20

Answer−20

Part (b) · 2 marks

Find the inverse function g⁻¹(x).

  1. 1y = 6 − 2x, swap: x = 6 − 2y
  2. 22y = 6 − x
  3. 3g⁻¹(x) = (6 − x)/2

Answer(6 − x)/2

Part (c) · 4 marks

Solve f(x) = g(2x − 7) for x.

  1. 1g(2x − 7) = 6 − 2(2x − 7) = 6 − 4x + 14
  2. 24x + 1 = 20 − 4x
  3. 38x = 19
  4. 4x = 19/8 = 2.375

Answer2.375

Part (d) · 2 marks

Find hh(2), applying h twice starting at 2.

  1. 1h(2) = 3^(2 − 2) = 3⁰ = 1
  2. 2hh(2) = h(1) = 3^(1 − 2) = 3⁻¹ = 1/3

Answer1/3 or 0.333…

Part (e) · 2 marks

Find x such that h⁻¹(x) = 10.

  1. 1h⁻¹(x) = 10 means x = h(10)
  2. 2x = 3^(10 − 2) = 3⁸ = 6561

Answer6561

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Also searched as: 0580/41/M/J/24 · 0580_s24_qp_41 · 0580 s24 41 · question 9.
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