Overview

Functional equations ask you to determine all functions satisfying a condition. Look for special inputs, symmetry, and invariants.

Key Ideas

• Try small values like $0, 1, -1$ when allowed.
• Substitute to eliminate variables and express everything in terms of $f(0)$.
• If the equation is symmetric, try swapping variables to get new relations.

Worked Example

If $f(x+1)=f(x)+2x+1$ and $f(0)=0$, find $f(3)$.

Compute step by step: $f(1)=0+1$, $f(2)=1+3=4$, $f(3)=4+5=9$. This matches $f(n)=n^2$ for nonnegative integers.

Practice Problems