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
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If f(x) = x + 1, then f(f(2)) equals:
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Answer: 4
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If f(x + y) = f(x) + f(y) for all x, y and f(1) = 3, then f(2) equals:
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Answer: 6
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If f(x) = 2x + 1, then f(3) equals:
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Answer: 7
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The identity function is defined by f(x) =
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Answer: x
Practice Questions
Practise RMO questions on Functional Equations. Answers are revealed after each question.
Start Practice →Frequently Asked Questions
What are the key concepts in Functional Equations for RMO?
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.
Which formulae are important for Functional Equations in RMO?
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.
How can I practise RMO Functional Equations questions for free?
Practise free RMO Functional Equations questions with instant answers and explanations on Safe Learning for Kids — no sign-up required.