Curriculum · Olympiad Topics · Functional Equations

Functional Equations

Substitution discipline, injectivity and surjectivity, and the Cauchy equation.

Objectives

Make systematic substitutions
Prove injectivity/surjectivity
Recognize Cauchy-type equations

The full interactive edition of this lesson ships in v1.5. The topic is live everywhere else — the problems below feed your Functional Equations mastery now.

Train the topic

●●●●○A function satisfies

f(x + y) = f(x) + f(y)

for all real

x, y

, and

f(1) = 3

. What is

f(7)

?Olympiad training ●●●●○A function satisfies

f(x) + 2f\!\left(\tfrac{1}{x}\right) = 3x

for all

x \ne 0

. What is

f(2)

?AMC 12 style