16. Consider the function $f$ defined by
$$
f(x)=1+\frac{2}{x} .
$$

Find the roots of the equation
$$
(\underbrace{f \circ f \circ \cdots \circ f}_{10 \text { times }})(x)=x,
$$
where "o" denotes composition of functions.
16. -1 and 2
Let $f^{(n)}(x)=(\underbrace{f \circ f \circ \cdots \circ f}_{n \text { times }})(x)$. For allowed values of $x$, note that $f^{(n)}(x)$ is of the form
$$
f^{(n)}(x)=\frac{a_{n} x+b_{n}}{c_{n} x+d_{n}}
$$
where $a_{n}, b_{n}, c_{n}, d_{n} \in \mathbb{Z}$ for all integers $n \geq 1$. The equation
$$
\frac{a_{n} x+b_{n}}{c_{n} x+d_{n}}=x \quad \text { or } \quad c_{n} x^{2}+\left(d_{n}-a_{n}\right) x-b_{n}=0
$$
has at most two real roots. Since $f(-1)=-1$ and $f(2)=2$, it follows that $f^{(n)}(-1)=-1$ and $f^{(n)}(2)=2$ for all $n \geq 1$. Thus, the roots of $f^{(10)}(x)=x$ are -1 and 2 .