Functional equation

A function f takes a number x and gives back one number, f(x). For example, if f(x) = 3x, then f(2) = 6. In a functional equation you are not told the rule. You are told a fact it obeys, and you must find f or some of its values.

The main tool is to put in clever values, such as x = 0, x = 1 or y = x. Each choice gives a new fact. You then combine the facts.

A small example

A function obeys f(x + y) = f(x) + f(y) for all numbers x and y, and f(1) = 3. Find f(0), f(2) and f(-1).

• Put x = y = 0: f(0) = f(0) + f(0). Take f(0) from both sides to get f(0) = 0.
• Put x = y = 1: f(2) = f(1) + f(1) = 6. In the same way f(3) = 9.
• Put x = 1 and y = -1: f(0) = f(1) + f(-1). So 0 = 3 + f(-1), and f(-1) = -3.

The pattern suggests f(n) = 3n for every integer n. A full answer must prove that, not just guess it. Proving the rule is what the harder questions ask for.

Where it fits

This goes beyond the Class 10 board syllabus. HBCSE's syllabus for the Mathematical Olympiad names it under algebra. The Mathematics Teachers' Association (India), MTA(I), conducts IOQM, the first stage of the Mathematical Olympiad Programme that HBCSE organises for NBHM.

About the author

Mohit Sardana

Chief Architect

An engineer (B.Tech, Computer Science) with over 20 years of preparing students for entrance exams, including as Dean at FIITJEE and Vice-President at Aakash. He is the chief architect of the TRUpreBoards evaluation engine.

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