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0580/43 Oct/Nov 2024, Question 11: Composite Functions

Cambridge IGCSE Mathematics 0580 · Paper 43 (Extended) · Calculator · 14 marks

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The question, in short

Use four given functions f, g, h and j to evaluate, combine, invert and simplify expressions, and solve equations involving them.

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

Part (a) · 1 mark

Evaluate function g at input −3.

  1. 1g(−3) = 1 − 2(−3)
  2. 2= 7

Answer7

Part (b) · 4 marks

Form the product of f and g, add the composite fg(x) and 1, then simplify fully.

  1. 1f(x)g(x) = (2x + 5)(1 − 2x) = 2x − 4x² + 5 − 10x
  2. 2fg(x) = 2(1 − 2x) + 5 = 7 − 4x
  3. 3Sum + 1 = −4x² − 8x + 5 + 7 − 4x + 1
  4. 4= −4x² − 12x + 13

Answer−4x² − 12x + 13

Part (c) · 2 marks

Determine the inverse function of g.

  1. 1x = 1 − 2y
  2. 22y = 1 − x
  3. 3g⁻¹(x) = (1 − x)/2

Answer(1 − x)/2

Part (d) · 2 marks

Compute h applied twice, starting at input 1.

  1. 1h(1) = 1/(1 + 1) = 1/2
  2. 2h(1/2) = 1/(1/2 + 1) = 1/(3/2) = 2/3

Answer2/3

Part (e) · 3 marks

Subtract h(x) from the reciprocal of f(x), giving one simplified fraction.

  1. 11/(2x + 5) − 1/(x + 1)
  2. 2= ((x + 1) − (2x + 5))/((2x + 5)(x + 1))
  3. 3= (−x − 4)/((2x + 5)(x + 1))

Answer(−x − 4)/((2x + 5)(x + 1)) or (−x − 4)/(2x² + 7x + 5)

Part (f) · 1 mark

Solve j(x) = 1/32 for x.

  1. 12ˣ = 1/32 = 2⁻⁵
  2. 2x = −5

Answer−5

Part (g) · 1 mark

Solve for x when the inverse of j equals 0.

  1. 1j⁻¹(x) = 0 means x = j(0)
  2. 2x = 2⁰ = 1

Answer1

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Also searched as: 0580/43/O/N/24 · 0580_w24_qp_43 · 0580 w24 43 · question 11.
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