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.

Easy Warm-up 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 $f(1/2)=1/2$.

Example 4: Injectivity Practice

If

for all real numbers $x,y$, your goal is to show that $f$ is injective.

Suppose $f(a)=f(b)$. Then

and

Since $f(a)=f(b)$, we obtain $a=b$.

Example 5: Surjectivity Practice

If

for all real numbers $x$, your job is to show that $f$ is surjective.

As $x$ varies over $\mathbb{R}$, the right-hand side attains every real value. Hence $f(f(x))$ attains every real value, so $f$ is surjective.

Example 6: What c makes the function 0?

Suppose

for all real numbers $x,y$.

Since $f$ is surjective, choose $c$ such that $f(c)=0$. Then

so $c=0$.

Example 7: Finding Solutions

Your job is to find all linear solutions to the following functional equation:

First, we let $f(x)=ax+b$.Then, substituting gives

Comparing coefficients yields

Since $a\neq 0$, we get $b=0$, so

Example 8: What is $f(0)$?

Suppose that

for all real numbers $x,y$.

Setting $x=y=0$ gives

hence

Example 9: Find the Functional Relation

Suppose

and $f(0)=0$.

Setting $y=x$ gives

so

Example 10: Iteration

If $f(f(x))=f(0)+x$, show that $f(f(f(f(x))))=2f(0)+x$.

Applying the given relation with $x$ replaced by $f(f(x))$ gives

Using the original equation again,

Example 11: Constant Solutions

Your goal here is to find all constant solutions of

Let $f(x)\equiv c$. Then

Thus

so the only constant solution is

Example 12: What is $f(0)$?

If

for all real numbers $x,y$, show that either $f(0)=1$ or $f$ is constant.

Setting $x=0$ gives

Letting $a=f(0)$, we obtain

Thus either $a=1$ or

Medium-Level Examples

Strategy Checklist

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

Practice Problems