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0580/41 May/Jun 2022, Question 4: Inverse Functions

Cambridge IGCSE Mathematics 0580 · Paper 41 (Extended) · Calculator · 11 marks

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The question, in short

Evaluate functions f, g, h, j at given inputs, invert g and j, solve equations, and combine f with h into one fraction.

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

Part (ai) · 1 mark

Substitute 2 into f(x) = 2x − 1 and compute the result.

  1. 1f(2) = 2 × 2 − 1 = 3

Answer3

Part (aii) · 1 mark

Apply g to the value of f(2), using g(x) = 3x − 2.

  1. 1gf(2) = g(3) = 3 × 3 − 2 = 7

Answer7

Part (b) · 2 marks

Rearrange g(x) = 3x − 2 to express x in terms of the output.

  1. 1y = 3x − 2, swap: x = 3y − 2
  2. 23y = x + 2
  3. 3g⁻¹(x) = (x + 2)/3

Answer(x + 2)/3

Part (c) · 2 marks

Set h(x) = 1/x equal to j(−2) with j(x) = 5ˣ, then solve for x.

  1. 1j(−2) = 5⁻² = 1/25
  2. 21/x = 1/25
  3. 3x = 25

Answer25

Part (d) · 2 marks

Subtract 1/x from 2x − 1 and combine into a single fraction.

  1. 1f(x) − h(x) = 2x − 1 − 1/x
  2. 2= (x(2x − 1) − 1)/x
  3. 3= (2x² − x − 1)/x

Answer(2x² − x − 1)/x

Part (e) · 1 mark

Apply j twice to 2, using j(x) = 5ˣ.

  1. 1j(2) = 5² = 25
  2. 2jj(2) = j(25) = 5²⁵ = 298 023 223 876 953 125
  3. 3≈ 2.98 × 10¹⁷

Answer2.98 × 10¹⁷ or 2.980... × 10¹⁷

Part (f) · 2 marks

Solve j⁻¹(x) = 4 by converting to j(4) with j(x) = 5ˣ.

  1. 1j⁻¹(x) = 4 means x = j(4).
  2. 2j(x) = 5ˣ, so x = 5⁴ = 625

Answer625

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Also searched as: 0580/41/M/J/22 · 0580_s22_qp_41 · 0580 s22 41 · question 4.
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