Lemmara logoLemmaraWorked solution

0580/43 May/Jun 2021, Question 10: Composite Functions

Cambridge IGCSE Mathematics 0580 · Paper 43 (Extended) · Calculator · 11 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Use the four given functions f, g, h and j to evaluate, invert, compose and differentiate, working with numbers and algebra throughout.

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

Part (ai) · 1 mark

Evaluate f at 2 using 3x − 2.

  1. 1f(2) = 3 × 2 − 2 = 4

Answer4

Part (aii) · 1 mark

Evaluate g at 2 using 5x − 7.

  1. 1g(2) = 5 × 2 − 7 = 3

Answer3

Part (aiii) · 1 mark

Evaluate the composite gf at 2.

  1. 1gf(2) = g(f(2)) = g(4)
  2. 2= 5 × 4 − 7 = 13

Answer13

Part (b) · 2 marks

Find the inverse function of f(x) = 3x − 2.

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

Answer(x+2)/3

Part (c) · 3 marks

Form the composite hf and expand to ax² + bx + c.

  1. 1hf(x) = (3x − 2)² + (3x − 2)
  2. 2(3x − 2)² = 9x² − 6x − 6x + 4 = 9x² − 12x + 4
  3. 3hf(x) = 9x² − 12x + 4 + 3x − 2 = 9x² − 9x + 2

Answer9x² − 9x + 2

Part (d) · 1 mark

Differentiate h(x) = x² + x.

  1. 1Multiply by the power, then take 1 off the power: axⁿ → naxⁿ⁻¹
  2. 2x² → 2x
  3. 3x → 1
  4. 4h′(x) = 2x + 1

Answer2x + 1

Part (ei) · 1 mark

Solve for x given j⁻¹(x) = 4 with j(x) = 3ˣ.

  1. 1j⁻¹(x) = 4 means x = j(4)
  2. 2x = 3⁴ = 81

Answer81

Part (eii) · 1 mark

Simplify the composite j⁻¹j(x).

  1. 1j⁻¹ undoes j, so j⁻¹j(x) = x

Answerx

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

Practise more composite functions questions

Real 0580 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on composite functions

Also searched as: 0580/43/M/J/21 · 0580_s21_qp_43 · 0580 s21 43 · question 10.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.