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0580/21 May/Jun 2025, Question 14: Algebraic Fractions

Cambridge IGCSE Mathematics 0580 · Paper 21 (Extended) · Non-calculator · 8 marks

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The question, in short

Given f(x) = 3x − 4 and g(x) = 4x + 1, evaluate f at −2, find the inverse of f, find a and b in fg(x) = ax + b, and combine 2/f(x) − 5/g(x) into one fraction.

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Part (a) · 1 mark

Work out f(−2) using f(x) = 3x − 4.

  1. 1f(−2) = 3 × (−2) − 4 = −10

Answer−10

Part (b) · 2 marks

Find the inverse function f⁻¹(x) for f(x) = 3x − 4.

  1. 1y = 3x − 4, so y + 4 = 3x
  2. 2x = (y + 4)/3
  3. 3f⁻¹(x) = (x + 4)/3

Answer(x + 4)/3

Part (c) · 2 marks

Given fg(x) = ax + b, find the values of a and b.

  1. 1fg(x) = 3(4x + 1) − 4
  2. 2= 12x + 3 − 4 = 12x − 1
  3. 3a = 12, b = −1

Answera = 12, b = −1

Part (d) · 3 marks

Simplify 2/f(x) − 5/g(x) to a single fraction in x.

  1. 12/(3x − 4) − 5/(4x + 1)
  2. 2= [2(4x + 1) − 5(3x − 4)]/((3x − 4)(4x + 1))
  3. 3= (8x + 2 − 15x + 20)/((3x − 4)(4x + 1))
  4. 4= (22 − 7x)/((3x − 4)(4x + 1))

Answer(22 − 7x)/((3x − 4)(4x + 1)) or (22 − 7x)/(12x² − 13x − 4)

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Also searched as: 0580/21/M/J/25 · 0580_s25_qp_21 · 0580 s25 21 · question 14.
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