G285 2021 Summer Problem Set Problem 2

Problem

Let $f(x,y) = \begin{cases}x^y & \text{ if } x^2>y \text{ and } |x|<y\\f(f(\sqrt{|x|},y),y) & \text{ otherwise} \end{cases}$ If $y$ is a positive integer, find the sum of all values of $x$ such that $f(x,y) \neq k$ for some constant $k$ .

$\textbf{(A)}\ -1 \qquad\textbf{(B)}\ -\frac{1}{2} \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ \frac{3}{8} \qquad\textbf{(E)}\ 1$

Solution

Note for $x \in \{ -1,1 \}$ , our function approaches an infinite recursion. Now, for $-1<x<1$ , we have the function approaches $-1$ , or $1$ , which also is an infinite recursion. The answer is $\boxed{\textbf{(C)}\ 0}$

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G285 2021 Summer Problem Set Problem 2

Problem

Solution