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4MA1/2HR Jan 2021, Question 17: Composite Functions

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 2HR (Higher) · Calculator · 9 marks

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The question, in short

Given f(x) = x² + 6 and g(x) = x − 10, work with these functions: evaluate a composite at 3, solve an equation, and handle h(x) = (2x − 4)/x including its domain and inverse.

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

Part (a) · 2 marks

Work out the value of fg(3), applying g first then f.

  1. 1g(3) = 3 − 10 = −7
  2. 2fg(3) = f(−7) = (−7)² + 6
  3. 3(−7)² = 49, so fg(3) = 49 + 6 = 55

Answer55

Part (b) · 3 marks

Find all x satisfying fg(x) = f(x), showing each algebraic step clearly.

  1. 1fg(x) = f(x − 10) = (x − 10)² + 6
  2. 2Set it equal to f(x): (x − 10)² + 6 = x² + 6
  3. 3Expand: x² − 20x + 100 + 6 = x² + 6
  4. 4The x² and the 6 cancel, leaving −20x + 100 = 0
  5. 5x = 5

Answer5

Part (c) · 1 mark

Give the x value excluded from h's domain.

  1. 1h(x) = (2x − 4)/x, and a fraction cannot have 0 on the bottom
  2. 2So x = 0 cannot be in the domain

Answer0

Part (d) · 3 marks

Find h⁻¹(x) for h(x) = (2x − 4)/x, written as h⁻¹(x) = …

  1. 1Write y = (2x − 4)/x
  2. 2Multiply by x: xy = 2x − 4
  3. 3Collect the x terms: xy − 2x = −4, so x(y − 2) = −4
  4. 4x = −4/(y − 2) = 4/(2 − y)
  5. 5Swapping the letters back, h⁻¹(x) = 4/(2 − x)

Answer4/(2 − x)

Stuck on a step? Read the free lesson: Composite Functions

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