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0580/42 Oct/Nov 2024, Question 9: Composite Functions

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 8 marks

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The question, in short

Given f(x) = 2x − 5 and g(x) = x² − 2x, evaluate f(7), the composite gf(7), the inverse f⁻¹(x), and simplify gf(x) − 3g(x) into the form ax² + bx + c.

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

Part (ai) · 1 mark

Work out f(7) using f(x) = 2x − 5.

  1. 1f(7) = 2 × 7 − 5 = 9

Answer9

Part (aii) · 1 mark

Find gf(7), applying g to the value of f(7).

  1. 1gf(7) = g(9) = 9² − 2 × 9 = 81 − 18 = 63

Answer63

Part (aiii) · 2 marks

Determine the inverse function f⁻¹(x) for f(x) = 2x − 5.

  1. 1y = 2x − 5, swap: x = 2y − 5
  2. 2y = (x + 5)/2, so f⁻¹(x) = (x + 5)/2

Answer(x + 5)/2

Part (b) · 4 marks

Find gf(x) − 3g(x) and express it as ax² + bx + c.

  1. 1gf(x) − 3g(x) = (2x − 5)² − 2(2x − 5) − 3(x² − 2x)
  2. 2= 4x² − 10x − 10x + 25 − 4x + 10 − 3x² + 6x
  3. 3= x² − 18x + 35

Answerx² − 18x + 35

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Also searched as: 0580/42/O/N/24 · 0580_w24_qp_42 · 0580 w24 42 · question 9.
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