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0580/23 Oct/Nov 2022, Question 19: Inverse Functions

Cambridge IGCSE Mathematics 0580 · Paper 23 (Extended) · Calculator · 6 marks

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The question, in short

Given f(x) = 5x − 3 for x > 1 and g(x) = 10/(x − 2) for x ≠ 2, find a composite function, an inverse, and a composition involving f and its inverse.

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

Part (a) · 2 marks

Find gf(x) and simplify the resulting expression.

  1. 1gf(x) = 10/((5x − 3) − 2)
  2. 2= 10/(5x − 5) = 2/(x − 1)

Answer2/(x − 1)

Part (b) · 3 marks

Find the inverse function g⁻¹(x).

  1. 1y = 10/(x − 2)
  2. 2x − 2 = 10/y
  3. 3x = 10/y + 2
  4. 4g⁻¹(x) = 10/x + 2

Answer10/x + 2 or (10 + 2x)/x

Part (c) · 1 mark

Find ff⁻¹(x − 1).

  1. 1f and f⁻¹ undo each other, so ff⁻¹(a) = a for any input a
  2. 2Here the input is x − 1, so ff⁻¹(x − 1) = x − 1

Answerx − 1

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Also searched as: 0580/23/O/N/22 · 0580_w22_qp_23 · 0580 w22 23 · question 19.
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