Algebra
# Functional Equations

Define the function $f^1(x) = \frac{1}{1-x}$, and $f^n (x) = f^1 ( f^{n-1} (x) )$ for positive integers $n$.

Evaluate $f^{36} ( 10 ) .$

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that

$5 f( x + y) + y^ 5 = f(x) + (x+y) ^ 5.$

Suppose that function $f$ satisfies

$f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2} \text{ and } f(0)=17.$

Which of the following represents the family of solutions for $f(x)$?

$k$ in the choices below is a constant.