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Functional Equations

Techniques for solving functional equations, substitution strategies, and olympiad-style problems

Key Concepts
  • Strategic substitutions reveal the structure of f
  • Injectivity and surjectivity narrow down the possibilities
  • Cauchy's functional equation and its standard solutions
  • Always verify that the candidate function actually works
Important Formulae
Cauchy's equation f(x+y) = f(x) + f(y) ⇒ f(x) = cx (with mild regularity)
Common substitutions Try x = y = 0, y = x, y = −x
Quick Tips
  • Test simple candidates first: f(x) = x, f(x) = c, f(x) = x².
  • Proving f is injective or monotonic often forces the answer.
Sample Practice Questions

  1. If f(x) = 2x + 1, then f(3) equals:

    • 6
    • 7
    • 8
    • 5
    Show answer

    Answer: 7

  2. If f(x) = x², then f(−2) equals:

    • −4
    • 4
    • 2
    • −2
    Show answer

    Answer: 4

  3. The identity function is defined by f(x) =

    • x
    • 1
    • 0
    • 2x
    Show answer

    Answer: x

  4. If f(x + y) = f(x) + f(y) for all x, y and f(1) = 3, then f(2) equals:

    • 3
    • 6
    • 9
    • 5
    Show answer

    Answer: 6

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Practice Questions

Practise RMO questions on Functional Equations. Answers are revealed after each question.

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Algebra