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\title{TJUSAMO Practice 13 - Functional Equations}
\author{PDiao06}
\date{\today}

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\maketitle

This lecture will be mostly problem based.  Functional equations = cleverness.

\section{Common Techniques}

Here is a list of things to remember to try to use with functional equations.
I took this straight from Reid Barton.  

\begin{enumerate}

\item Guess the answers and use them.
\item Plug in $0$ and $1$ or other small cases.  
\item Make things equal.  This could be as simple as $x=y$ or other much less
obvious stuff.
\item Repeated application.  Applying the function to the whole thing again.
\item Do a lot of casework based on certain values.  
\item The idea of a surjective or injective function can force a certain value.
\item Build up a lot of values, show they're dense, and then use continuity.

\end{enumerate}

\section{Practice Problems}

These are all compiled from different MOSP sources.

\begin{enumerate}

\item (Cauchy's Functional Equation) Find all continuous solutions to $f(x+y) =
f(x) + f(y)$.  
\item Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the
equality $f(f(x) + y) = f(x^2-y) + 4f(x)y$ holds for all $x,y \in \mathbb{R}$.
\item Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for 
all real numbers $x,y,z,t$
$$(f(x) + f(z))(f(y) + f(t)) = f(xy-zt) + f(xt + yz)$$
\item Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ which satify 
$f(m+f(n)) = f(f(m)) + f(n)$ for all $m,n\in \mathbb{Z}$.
\item Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that 
$f(x+y) = f(x) + f(y)$ and $f(xy) = f(x)f(y)$ for all $x,y \in \mathbb{R}$.
\item Let $f: [0,1] \rightarrow \mathbb{R}$ be a function such that $f(1)=1$,
$f(x) \ge 0$, and if $x,y, x+y$ all lie in $[0,1]$, then $f(x+y) \ge f(x)+ f(y)$
.  Prove that $f(x)\le2x$ for all $x \in [0,1]$.
\item Find all functions $f: \mathbb{R} \rightarrow [0,\infty)$ such that for
all $x,y$ real:
$$f(x^2+y^2) = f(x^2-y^2)+f(2xy)$$
\item Find all polynomials $p(x)$ such that for all $x$:
$$(x-16)p(2x)=16(x-1)p(x)$$
\item Determine all polynomials $P(x)$ with real coefficients such that
$$(x^3 + 3x^2 + 3x + 2)P(x-1)=(x^3-3x^2+3x-2)P(x)$$
\item Find all pairs of functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and
$g: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
$$f(x+g(y)) = xf(y) - yf(x) + g(x)$$
\item Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that 
for all $x,y \in \mathbb{R}$,
$$f(x-f(y)) = f(f(y)) + xf(y) + f(x) -1$$

\end{enumerate}

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