Overview

Advanced functional equations often mix algebra and number theory. Expect to find hidden constraints via special inputs and symmetry.

Key Ideas

• If $f$ is defined on integers, parity and modular arguments can restrict it.
• Look for fixed points: solve $f(x)=x$ or $f(x)=c$.
• Iteration (apply the equation repeatedly) can uncover structure.

Core Skills

Pin Down Constants

Use $x=0$, $y=0$, $x=y$, or $x=-y$ to extract $f(0)$ and other constants.

Iterate the Functional Equation

Apply the equation multiple times to relate $f(x)$ to $f(f(x))$ or $f(x+c)$.

Use Injectivity/Surjectivity

Prove injectivity by showing $f(a)=f(b)\Rightarrow a=b$, then use it to simplify equations.

Worked Example

If $f(x)=2x$ satisfies $f(x+y)=f(x)+f(y)$ for all $x,y$, then $f$ is additive. Many contest problems ask you to prove that linear functions are the only solutions.

More Examples

Example 1: Fixed Point

If $f(f(x))=x$, show $f$ is bijective.

Apply $f$ to both sides to get $f(x)=f(f(f(x)))$, implying surjectivity and injectivity.

Example 2: Integer Restriction

If $f:\mathbb{Z}\to\mathbb{Z}$ and $f(x+1)=f(x)+2$, then $f(x)=2x+c$.

Example 3: Symmetry

If $f(x)+f(1-x)=1$, then setting $x=1-x$ gives $f(1/2)=1/2$.

Strategy Checklist

• Plug in special values to find constants.
• Iterate the equation to uncover structure.
• Use injectivity or surjectivity when possible.

Practice Problems