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0580/42 Feb/Mar 2023, Question 11: Algebraic Fractions

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 10 marks

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The question, in short

Use four given functions f, g, h and j to evaluate, solve, compose and identify a self-inverse function.

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

Part (a) · 1 mark

Evaluate j at x = −1 using j(x) = x².

  1. 1j(−1) = (−1)²
  2. 2= 1

Answer1

Part (b) · 2 marks

Solve f(x) + g(x) = 0 for x with the two linear functions.

  1. 12x − 1 + 3x + 2 = 0
  2. 25x + 1 = 0
  3. 3x = −1/5 = −0.2

Answer−1/5 or −0.2

Part (c) · 2 marks

Form the composite gg(x) and simplify the resulting expression.

  1. 1gg(x) = g(3x + 2) = 3(3x + 2) + 2
  2. 2= 9x + 6 + 2 = 9x + 8

Answer9x + 8

Part (d) · 4 marks

Combine hf(x) and gh(x) into one simplified single fraction.

  1. 1hf(x) = 1/(2x − 1) and gh(x) = 3/x + 2
  2. 2Common denominator x(2x − 1)
  3. 3= [x + 3(2x − 1) + 2x(2x − 1)]/(x(2x − 1))
  4. 4= (x + 6x − 3 + 4x² − 2x)/(x(2x − 1))
  5. 5= (4x² + 5x − 3)/(x(2x − 1))

Answer(4x² + 5x − 3)/(x(2x − 1))

Part (e) · 1 mark

Choose which of f, g, h, j equals its own inverse.

  1. 1hh(x) = h(1/x) = 1/(1/x) = x
  2. 2So h⁻¹(x) = h(x): ring h(x) = 1/x

Answerh(x)

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