AQA GCSE Mathematics Paper 2 (Higher), June 2025: Question 25

4 marks · Hard difficulty · Proof

Prove algebraically that fg(x) - a f^(-1)(x) is always a multiple of 3 given f(x) = ax + b and g(x) = (x + b)/a.

Practise this question

Question

Question 25: f(x) = ax + b and g(x) = (x + b)/a, where a and b are positive integers. Prove that fg(x) - a f^(-1)(x) is always a multiple of 3. The question is worth 4 marks.

Mark scheme

Show the mark scheme Mark scheme for Question 25: B1 for finding fg(x) as a((x + b)/a) + b or x + 2b. B1 for finding f^(-1)(x) as (x - b)/a. B1 for substituting both into the expression to give a((x + b)/a) + b - a((x - b)/a) or x + 2b - (x - b). B1 for fully correct simplification leading to 3b with no errors.

How to answer it

Algebraic Proof with Composite and Inverse Functions

📋 What this question tests

This Grade 8/9 question tests your ability to find a composite function, derive an inverse function algebraically, handle subtraction with algebraic brackets, and simplify terms to complete a formal algebraic proof.

Question 25 (4 Marks)

Given f(x) = ax + b and g(x) = (x + b)/a, where a and b are positive integers. Prove that fg(x) − af⁻¹(x) is always a multiple of 3.

📐 Step-by-Step Full Solution

  1. Find fg(x): Substitute g(x) into f(x).
    fg(x) = f(g(x)) = a[(x + b)/a] + b
    The factor of a cancels with the denominator a :
    fg(x) = (x + b) + b = x + 2b
  2. Find f⁻¹(x): Rearrange y = ax + b to make x the subject.
    y − b = ax ⇒ x = (y − b)/a
    Therefore, f⁻¹(x) = (x − b)/a
  3. Substitute both into the given expression:
    fg(x) − a f⁻¹(x) = (x + 2b) − a[(x − b)/a]
  4. Simplify the expression carefully:
    The factor of a cancels:
    = (x + 2b) − (x − b)
    Expand the brackets (remember the double negative: −(−b) = +b):
    = x + 2b − x + b
    = 3b
  5. Conclusion:
    Since b is a positive integer, 3b must always be a multiple of 3.

✅ Mark Scheme Breakdown

  • B1: Correct expression for fg(x), e.g. a((x + b)/a) + b or x + 2b .
  • B1: Correct expression for f⁻¹(x), e.g. (x − b)/a .
  • B1: Fully correct expression in terms of x, e.g. x + 2b − (x − b) or x + 2b − x + b .
  • B1: Reaching the final result of 3b with completely sound working and no algebraic errors.
Note: The examiner explicitly states: "Ignore explanation about why 3b is a multiple of 3." Simply reaching an error-free 3b secures full marks!

💡 Key Knowledge

  • Composite Functions: fg(x) means apply g first, then substitute the result everywhere you see x in f .
  • Inverse Functions: To find f⁻¹(x) , set y = f(x) , swap or rearrange to make x the subject, then write in terms of x .
  • Multiples in Proofs: An algebraic expression is a multiple of 3 if it can be written as 3 × (integer) .

🧠 Exam Technique

  • Always use brackets after a minus sign: Writing − (x − b) prevents the fatal error of forgetting to change the sign to + b .
  • Never substitute numbers: The mark scheme specifically states: "Substitution of values for letters with no further correct work: Zero." Proof requires general algebra, not examples.
  • Break into separate steps: Work out fg(x) on its own first, then f⁻¹(x) , then combine them. This guarantees method marks even if your final expansion goes wrong.

❌ Common Traps & Pitfalls

  • The Sign Error Trap: Writing x + 2b − x − b = b instead of distributed signs. The mark scheme explicitly highlights: x + 2b − x − b = 3b (error) → B0 . You cannot "fudge" your way to 3b!
  • Missing Brackets: Writing x + 2b − x − b and then writing = 3b loses both the 3rd and 4th marks because recovery from missing brackets is not allowed.
  • Confusing fg(x) with gf(x): Working out g(f(x)) instead of f(g(x)) will forfeit the first mark and disrupt the whole proof.

Topics

Algebra · 3.2.1 Notation, vocabulary and manipulation

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 2 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.