Overview

AMC and AIME functional equations usually test algebraic manipulation and strategic substitution rather than heavy theory.

Key Ideas

• Plug in special values like $x=0$, $y=0$, $y=x$, and $y=-x$.
• Check linear candidates: $f(x)=cx$, $f(x)=x+c$, $f(x)=cx+d$.
• Use symmetry in the equation to deduce parity or linearity.

Core Skills

Use Strategic Substitution

Plugging in $0$, $1$, $x$, $-x$, or $y=-x$ often isolates constants or shows that $f$ is even/odd.

Test Simple Families

Try constant, linear, or multiplicative candidates. Many contest problems are designed so a small family works.

Build a Table of Values

Once $f(0)$ or $f(1)$ is found, use the equation to compute $f(2)$, $f(3)$, and detect a pattern.

Worked Example

Find all $f$ such that $f(x+y)=f(x)+f(y)$ for all real $x,y$.

Set $y=0$ to get $f(0)=0$. Then $f(nx)=nf(x)$ for integers $n$ by induction. On contest problems, continuity or monotonicity is often implied, giving $f(x)=cx$ for some constant $c$.

More Examples

Example 1: Constant Solution Check

Solve $f(x+y)=f(x)f(y)$ with $f(0)=2$.

Plug $x=y=0$: $f(0)=f(0)^2$ gives $2=4$, impossible. So no such function.

Example 2: Odd Function

If $f(x)+f(-x)=0$ for all $x$, what does this say about $f$?

The function is odd: $f(-x)=-f(x)$.

Example 3: Linear Candidate

Solve $f(x+y)=f(x)+f(y)+2$ for all $x,y$.

Try $f(x)=ax+b$. Then $ax+ay+b = ax+b+ay+b+2$ gives $b=-2$. So $f(x)=ax-2$.

Strategy Checklist

• Plug in $0$, $x$, and $-x$ first.
• Test constant or linear candidates.
• Track parity (even/odd) when $f(x)$ and $f(-x)$ appear.
• Verify solutions in the original equation.

Common Pitfalls

• Assuming continuity without checking the problem statement.
• Forgetting to verify candidate functions.
• Overlooking constant solutions.

Practice Problems