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0580/41 May/Jun 2021, Question 12: Composite Functions

Cambridge IGCSE Mathematics 0580 · Paper 41 (Extended) · Calculator · 14 marks

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The question, in short

Using f(x) = 3 − 2x, g(x) = x² + 5 and h(x) = x³, evaluate, combine, solve and invert these functions as asked in each part.

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

Part (a) · 1 mark

Work out the value of f(−5).

  1. 1f(−5) = 3 − 2 × (−5) = 3 + 10 = 13

Answer13

Part (b) · 2 marks

Find ff(x), giving the result in its simplest form.

  1. 1ff(x) = 3 − 2(3 − 2x)
  2. 2= 3 − 6 + 4x = 4x − 3

Answer4x − 3

Part (c) · 4 marks

Solve the equation g(x) = f(x) + 37.

  1. 1x² + 5 = 3 − 2x + 37
  2. 2x² + 2x − 35 = 0
  3. 3(x + 7)(x − 5) = 0
  4. 4x = −7 or x = 5

Answerx = −7 or x = 5

Part (d) · 2 marks

Find the inverse function f⁻¹(x).

  1. 1y = 3 − 2x, so 2x = 3 − y
  2. 2x = (3 − y)/2
  3. 3f⁻¹(x) = (3 − x)/2

Answer(3 − x)/2

Part (e) · 5 marks

Find hf(x) + g(x), giving the answer in its simplest form.

  1. 1hf(x) + g(x) = (3 − 2x)³ + x² + 5
  2. 2(3 − 2x)² = 9 − 12x + 4x²
  3. 3(3 − 2x)³ = (9 − 12x + 4x²)(3 − 2x) = 27 − 54x + 36x² − 8x³
  4. 4Total = 27 − 54x + 36x² − 8x³ + x² + 5 = 32 − 54x + 37x² − 8x³

Answer32 − 54x + 37x² − 8x³

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Also searched as: 0580/41/M/J/21 · 0580_s21_qp_41 · 0580 s21 41 · question 12.
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