50. How many continuous real-valued functions f are there with domain [-1,1] such that (f(x))^2 = x^2 for each x in [-1,1] A) One B) Two C) Three D) Four E) Infinite (Correct Answer D) Since f(x)^2 = x^2 we know f(x) = +/-x for every x in [-1,1] My first guess was that there were two continuous functions namely f(x) = x and f(x) = -x, I'm unsure how they constructed two more and was wondering if someone could explain the answer to me in more detail. My only conclusion was that they used |x| and -|x| but I was under the impression that these are not continuous, I may just be confusing differentiable with continuity however. 56. For every set S and every metric d on S, which of the following is a metric on S? A) 4 + d B) e^d - 1 C) d - |d| D) d^2 E) Root(d) (Correct Answer E) 4 + d is incorrect since 4 + d(x,x) != 0 e^d - 1 is incorrect since it fails the triangle inequality. EX: S=Z d(x,y) = |y-x| d(0,1) + d(1,2) >/= d(0,2) but under e^d-1 this is the ienquality~ 2e - 2 >/= e^2 -1 which is inconsisent C) Is 0 for everything which is not a metric on any set with more then 1 element. D/E) I thought both of these were metrics could someone please clarify why d^2 fails to be one? 61. What is the greatest integer that divides p^4-1 for every prime number p greater then 5? A) 12 B) 30 C) 48 D) 120 E) 240 (Correct Answer E) I had no idea how to start this question other then plugging in 6 finding a prime factorization and then comparing it with the other integers to see if I could eliminate possibilities (I highly doubt this is the correct way to go about this problem).