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0580/43 Oct/Nov 2021, Question 11: Composite Functions

Cambridge IGCSE Mathematics 0580 · Paper 43 (Extended) · Calculator · 11 marks

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The question, in short

Use the four given functions f, g, h and j to evaluate, solve and invert, working with powers and composite functions.

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

Part (ai) · 1 mark

Evaluate the exponential function h at input 3.

  1. 1h(3) = 4³ = 64

Answer64

Part (aii) · 1 mark

Evaluate the composite fh at input 3.

  1. 1fh(3) = f(64) = 2 × 64 − 1 = 127

Answer127

Part (b) · 4 marks

Solve gf(x) = 0 by substituting f into g and solving the resulting quadratic.

  1. 1gf(x) = (2x − 1)² + 2(2x − 1)
  2. 2= 4x² − 2x − 2x + 1 + 4x − 2 = 4x² − 1
  3. 34x² − 1 = 0, so (2x − 1)(2x + 1) = 0
  4. 4x = ±1/2

Answer±1/2

Part (c) · 2 marks

Given that the inverse of p equals f, determine p(x).

  1. 1p is the inverse of f: y = 2x − 1
  2. 2y + 1 = 2x, so x = (y + 1)/2
  3. 3p(x) = (x + 1)/2

Answer(x+1)/2

Part (d) · 3 marks

Solve h(x)j(x) = 1/√2 by combining the exponentials and equating indices.

  1. 14ˣ × 2ˣ = 2²ˣ × 2ˣ = 2³ˣ
  2. 21/√2 = 2^(−1/2)
  3. 33x = −1/2
  4. 4x = −1/6

Answer−1/6

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Also searched as: 0580/43/O/N/21 · 0580_w21_qp_43 · 0580 w21 43 · question 11.
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