2011 AMC 12A Problems/Problem 21

Problem

Let $f_{1}(x)=\sqrt{1-x}$ , and for integers $n \geq 2$ , let $f_{n}(x)=f_{n-1}(\sqrt{n^2 - x})$ . If $N$ is the largest value of $n$ for which the domain of $f_{n}$ is nonempty, the domain of $f_{N}$ is $[c]$ . What is $N+c$ ?

$\textbf{(A)}\ -226 \qquad \textbf{(B)}\ -144 \qquad \textbf{(C)}\ -20 \qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ 144$

Solution

The domain of $f_{1}(x)=\sqrt{1-x}$ is defined when $x\leq-1$ . $f_{2}(x)=f_{1}(\sqrt{4-x})=\sqrt{1-\sqrt{4-x}}$ . Applying the domain of $f_{1}(x)$ and the fact that square roots must be positive, we get $0\leq\sqrt{4-x}\leq1$ . Simplify this to arrive at the domain for $f_{2}(x)$ , which is defined when $3\leq x\leq4$ . Repeat this process for $f_{3}(x)=\sqrt{1-\sqrt{4-\sqrt{9-x}}}$ to get a domain of $-7\leq x\leq0$ . For $f_{4}(x)$ , since square roots are positive, we can exclude the negative values of the previous domain to arrive at $\sqrt{16-x}=0$ as the domain of $f_{4}(x)$ . We now arrive at a domain with a single number that defines $x$ , however, since we are looking for the largest value for $n$ for which the domain of $f_{n}$ is nonempty, we must continue until we arrive at a domain that is empty. We continue with $f_{5}(x)$ to get a domain of $\sqrt{25-x}=16$ . Solve for $x$ to get $x=-231$ . Since square roots cannot be negative, this is the last nonempty domain. We add to get $5-231=\boxed{\textbf{(A)}\ -226}$ .

2011 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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2011 AMC 12A Problems/Problem 21

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