For each pair of functions determine if f(n) ? ?(g(n)) or f(n) ? ?(g(n)) or f(n)...
For each pair of functions determine if f(n) ? ?(g(n)) or f(n) ? ?(g(n)) or f(n) ? O(g(n)) and provide a proof as specified.
For each of the following, give a proof using the definitions.
1. f(n) = log(n), g(n) = log(n + 1)
2. f(n) = n3 + nlog(n) ? n, g(n) = n4 + n
3. f(n) = log(n!), g(n) = nlog(n)
4. f(n) = log3(n), g(n) = log2(n)
5. f(n) = log(n), g(n) = log(log(n))
Homework Answers
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For each pair of functions determine if f(n) ? ?(g(n)) or f(n) ?
?(g(n)) or f(n)...
For each pair of functions f(n) and g(n), indicate whether f(n) = O(g(n)), f(n) = Ω(g(n)), and/or...
For each pair of functions f(n) and g(n), indicate whether f(n) = O(g(n)), f(n) = Ω(g(n)), and/or f(n) = Θ(g(n)), and provide a brief explanation of your reasoning. (Your explanation can be the same for all three; for example, “the two functions differ by only a multiplicative constant” could justify why f(n) = n, g(n) = 2n are related by big-O, big-Omega, and big-Theta.) i. f(n) = n^2 log n, g(n) = 100n^2 ii. f(n) = 100, g(n) = log(log(log...
Compare the following pairs of functions f, g. In each case, say whether f- o(g) f-w(g), or f = Θ...
Compare the following pairs of functions f, g. In each case, say whether f- o(g) f-w(g), or f = Θ(g), and prove your claim. 157. f(n) -100n+logn, gn) (logn)2. 158,介f(n) = logn, g(n) = log log(n2). 159. . f(n)-n2/log n, g(n) = n(log n)2. 160·介介f(n)-(log n)106.9(n)-n10-6 . 161. (n)logn, g(n) (log nlog n 162. f(n) n2, gn) 3.
Compare the following pairs of functions f, g. In each case, say whether f- o(g) f-w(g), or f = Θ(g), and prove...
1. a) Let f(n) = 6n2 - 100n + 44 and g(n) = 0.5n3 . Prove...
1. a) Let f(n) = 6n2 - 100n + 44 and g(n) =
0.5n3 . Prove that f(n) = O(g(n)) using the definition
of Big-O notation. (You need to find constants c and n0).
b) Let f(n) = 3n2 + n and g(n) = 2n2 . Use
the definition of big-O notation to prove that
f(n) = O(g(n)) (you need to find constants c and n0) and
g(n) = O(f(n)) (you need to find constants c and n0).
Conclude that...
(g(n)), g(n) is (f(n)), or both. 1. For each of the following pairs of functions determine...
(g(n)), g(n) is (f(n)), or both. 1. For each of the following pairs of functions determine if f(n) is f(n) = (na – n)/2, g(n) = 3n · f(n) = n log n, g(n) = n/n/2
3. (10 pts) For each of the following functions f(n), prove the stated claim by providing...
3. (10 pts) For each of the following functions f(n), prove the stated claim by providing constants no C1, and c2 such that for all n2 no, cig(n) S f(n) or f(n) c2g(n), and provide a calculation that shows that this inequality does indeed hold (a) f(n) 2n2 3n3-50nlgn10 0(n3) O(g(n)) (b) f(n)-2n log n + 3n2-10n-10-Ω ( 2)-0(g(n))
1. For each of the following pairs of functions, prove that f(n)-O(g(n)), and / or that...
1. For each of the following pairs of functions, prove that f(n)-O(g(n)), and / or that g(n) O(f(n)), or explain why one or the other is not true. (a) 2"+1 vs 2 (b) 22n vs 2" VS (c) 4" vs 22n (d) 2" vs 4" (e) loga n vs log, n - where a and b are constants greater than 1. Show that you understand why this restriction on a and b was given. f) log(0(1) n) vs log n....
Find f(x)) and g(f(x)) and determine whether the pair of functions f and g are inverses...
Find f(x)) and g(f(x)) and determine whether the pair of functions f and g are inverses of each other. f(x) = 9x +3 and g(x)= a. f(g(x)) = b. gcf(x) = (Simplify your answer.) (Simplify your answer.) o f and g are inverses of each other. fand g are not inverses of each other. O
(x)). For each pair of functions f and g below, find f(g(x)) and g Then, determine...
(x)). For each pair of functions f and g below, find f(g(x)) and g Then, determine whether f and g are inverses of each other. Simplify your answers as much as possible. (Assume that your expressions are defined for all x in the domain of the composition. You do not have to indicate the domain.) (a) f(x) = x + 4 (b) f(x) = - -, 0 3x x 5 ? g(x) = x - 4 f(g(x)) = 0 8(x)...
For each pair of functions f and g below, find f(g(x)) and g(x)). Then, determine whether...
For each pair of functions f and g below, find f(g(x)) and g(x)). Then, determine whether f and g are inverses of each other. Simplify your answers as much as possible. (Assume that your expressions are defined for all x in the domain of the composition. You do not have to indicate the domain.) (a) f(x) = -,x0 (b) f(x) = x + 4 $(x) = -,x+0 x 5 ? g(x) = -x + 4 $($(x)) = 0 (g(x)) =...
4. Determine whether or not the following are true and provide a full derivation explaining your ...
4. Determine whether or not the following are true and provide a full derivation explaining your answer for each. The domain of the functions of n below is the positive real numbers. For convenience, you may assume that the logs are in the base of your choice, but you should specify what base you are using in your derivation. 510g(n+is O(n) d. +3 log(log(n))+ Slozat)+ is o 5log(n+1) e. log(n3 n-3) is 0(n2)
4. Determine whether or not the following...
Compare the following pairs of functions f, g. In each case, say whether f- o(g) f-w(g), or f = Θ(g), and prove your claim. 157. f(n) -100n+logn, gn) (logn)2. 158,介f(n) = logn, g(n) = log log(n2). 159. . f(n)-n2/log n, g(n) = n(log n)2. 160·介介f(n)-(log n)106.9(n)-n10-6 . 161. (n)logn, g(n) (log nlog n 162. f(n) n2, gn) 3.
Compare the following pairs of functions f, g. In each case, say whether f- o(g) f-w(g), or f = Θ(g), and prove...
1. a) Let f(n) = 6n2 - 100n + 44 and g(n) =
0.5n3 . Prove that f(n) = O(g(n)) using the definition
of Big-O notation. (You need to find constants c and n0).
b) Let f(n) = 3n2 + n and g(n) = 2n2 . Use
the definition of big-O notation to prove that
f(n) = O(g(n)) (you need to find constants c and n0) and
g(n) = O(f(n)) (you need to find constants c and n0).
Conclude that...
(g(n)), g(n) is (f(n)), or both. 1. For each of the following pairs of functions determine if f(n) is f(n) = (na – n)/2, g(n) = 3n · f(n) = n log n, g(n) = n/n/2
3. (10 pts) For each of the following functions f(n), prove the stated claim by providing constants no C1, and c2 such that for all n2 no, cig(n) S f(n) or f(n) c2g(n), and provide a calculation that shows that this inequality does indeed hold (a) f(n) 2n2 3n3-50nlgn10 0(n3) O(g(n)) (b) f(n)-2n log n + 3n2-10n-10-Ω ( 2)-0(g(n))
1. For each of the following pairs of functions, prove that f(n)-O(g(n)), and / or that g(n) O(f(n)), or explain why one or the other is not true. (a) 2"+1 vs 2 (b) 22n vs 2" VS (c) 4" vs 22n (d) 2" vs 4" (e) loga n vs log, n - where a and b are constants greater than 1. Show that you understand why this restriction on a and b was given. f) log(0(1) n) vs log n....
Find f(x)) and g(f(x)) and determine whether the pair of functions f and g are inverses of each other. f(x) = 9x +3 and g(x)= a. f(g(x)) = b. gcf(x) = (Simplify your answer.) (Simplify your answer.) o f and g are inverses of each other. fand g are not inverses of each other. O
(x)). For each pair of functions f and g below, find f(g(x)) and g Then, determine whether f and g are inverses of each other. Simplify your answers as much as possible. (Assume that your expressions are defined for all x in the domain of the composition. You do not have to indicate the domain.) (a) f(x) = x + 4 (b) f(x) = - -, 0 3x x 5 ? g(x) = x - 4 f(g(x)) = 0 8(x)...
For each pair of functions f and g below, find f(g(x)) and g(x)). Then, determine whether f and g are inverses of each other. Simplify your answers as much as possible. (Assume that your expressions are defined for all x in the domain of the composition. You do not have to indicate the domain.) (a) f(x) = -,x0 (b) f(x) = x + 4 $(x) = -,x+0 x 5 ? g(x) = -x + 4 $($(x)) = 0 (g(x)) =...
4. Determine whether or not the following are true and provide a full derivation explaining your answer for each. The domain of the functions of n below is the positive real numbers. For convenience, you may assume that the logs are in the base of your choice, but you should specify what base you are using in your derivation. 510g(n+is O(n) d. +3 log(log(n))+ Slozat)+ is o 5log(n+1) e. log(n3 n-3) is 0(n2)
4. Determine whether or not the following...
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