Lemmara logoLemmaraWorked solution

0580/43 Oct/Nov 2025, Question 19: Composite Functions

Cambridge IGCSE Mathematics 0580 · Paper 43 (Extended) · Calculator · 8 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Given f(x) = 5ˣ, g(x) = 3x − 2 and h(x) = x² + 1, evaluate, invert and compose these functions to answer five parts.

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

Part (a) · 1 mark

Work out f(5) using the definition of f.

  1. 1f(5) = 5⁵ = 3125

Answer3125

Part (b) · 1 mark

Substitute 8x into g and simplify the resulting expression.

  1. 1g(8x) = 3(8x) − 2
  2. 2= 24x − 2

Answer24x − 2 or 2(12x − 1)

Part (c) · 2 marks

Find the inverse function g⁻¹(x) by rearranging g.

  1. 1y = 3x − 2
  2. 2Swap: x = 3y − 2
  3. 3y = (x + 2)/3
  4. 4g⁻¹(x) = (x + 2)/3

Answer(x + 2)/3 or x/3 + 2/3

Part (d) · 3 marks

Solve gh(x) = 364 for the positive value of x.

  1. 1gh(x) = 3(x² + 1) − 2 = 3x² + 1
  2. 23x² + 1 = 364
  3. 3x² = 121
  4. 4x = 11

Answer11

Part (e) · 1 mark

Evaluate ff⁻¹(12) using the relationship between a function and its inverse.

  1. 1f and f⁻¹ undo each other
  2. 2ff⁻¹(12) = 12

Answer12

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/O/N/25 · 0580_w25_qp_43 · 0580 w25 43 · question 19.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.