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moo5003
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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).
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).