Problem:
Let f1​(x)=1−x​, and for integers n≥2, let fn​(x)=fn−1​(n2−x​). If N is the largest value of n for which the domain of fn​ is nonempty, the domain of fN​ is {c}. What is N+c ?
Answer Choices:
A. −226
B. −144
C. −20
D. 20
E. 144 Solution:
Because f2​(x)=1−4−x​​,f2​(x) is defined if and only if 0≤4−x​≤1, so the domain of f2​ is the interval [3,4]. Similarly, the\
domain of f3​ is the solution set of the inequality 3≤9−x​≤4, which is the interval [−7,0], and the domain of f4​ is the solution set of the inequality −7≤16−x​≤0, which is {16}. The domain of f5​ is the solution set of the equation 25−x​=16, which is {−231}, and because the equation 36−x​=−231 has no real solutions, the domain of f6​ is empty. Therefore N+c=5+(−231)=−226​ .