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0580/42 Oct/Nov 2023, Question 11: Composite Functions

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 14 marks

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The question, in short

Use three given functions to evaluate, compose, invert and combine them, and to solve for unknown coefficients and an index.

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

Part (a) · 1 mark

Evaluate the squared function g at input 3.

  1. 1g(3) = (3 − 1)² = 4

Answer4

Part (b) · 2 marks

Substitute x − 2 into the linear function f and simplify the result.

  1. 1f(x − 2) = 1 − 3(x − 2) = 1 − 3x + 6
  2. 2= 7 − 3x

Answer7 − 3x

Part (c) · 2 marks

Determine the inverse function of the linear function f.

  1. 1Write y = 1 − 3x, then swap x and y: x = 1 − 3y
  2. 2Rearrange: 3y = 1 − x
  3. 3Divide by 3: f⁻¹(x) = (1 − x)/3

Answer(1 − x)/3

Part (d) · 5 marks

Expand the difference of two composite products and match coefficients to find a, b, c.

  1. 1gf(x) = (1 − 3x − 1)² = 9x²
  2. 2g(x)f(x) = (x² − 2x + 1)(1 − 3x) = −3x³ + 7x² − 5x + 1
  3. 3gf(x) − g(x)f(x) = 9x² + 3x³ − 7x² + 5x − 1 = 3x³ + 2x² + 5x − 1
  4. 4a = 2, b = 5, c = −1

Answera = 2, b = 5, c = −1

Part (e) · 3 marks

Subtract the linear function from the reciprocal function, giving one simplified fraction.

  1. 13/x − (1 − 3x) = (3 − x(1 − 3x))/x
  2. 2= (3 − x + 3x²)/x

Answer(3 − x + 3x²)/x

Part (f) · 1 mark

Substitute xⁿ into the reciprocal function and equate exponents to find n.

  1. 13/xⁿ = 3x⁻ⁿ = 3x⁷
  2. 2−n = 7, n = −7

Answer−7

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Also searched as: 0580/42/O/N/23 · 0580_w23_qp_42 · 0580 w23 42 · question 11.
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