# Growth of Functions

Need solutions for the following questions with step by step work shown
(a)
Let $f(x) = x^2 +x + 15$ and $g(x) = x^2 log(x) + 10$. Prove that f(x) is O(g(x)) by:
1. Providing witnesses C and k and
2. Proving that the inequality holds for the value you choose in 1.

(b)
Let $f(x) = x^5 + 10$ and $g(x) = x^5 + x + 10$
1. Prove that $f is O(g)$
2. Prove that $f is Ω(g)$
3. Given parts a and b, what other relationship can you show about f and g?

• Jack Smith
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Please provide full answers, and please solve them in a way that a someone who semi-understands (me) can comprehend and study it in the future

• Erdos
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That's what Matchmaticins actually expects from the answerers. All solutions should be written so someone with basic background can understand. I wrote a very detailed solution, but let me know if you need any clarification.

Answers can be viewed only if
1. The questioner was satisfied and accepted the answer, or
2. The answer was disputed, but the judge evaluated it as 100% correct.

1 Attachment

• Erdos
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• Jack Smith
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thanks phil. Can you please label parts 1 and 2 of (a) and also if possible solve the question with some other relatively lower value for C (say 3,4,5,17) .

• Jack Smith
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Will really appreciate if you'd write back:)

• Erdos
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What I have written for (a) answers part 2 of the question. The last line answers part 1, i.e. k=1 and C=e^2.

• Erdos
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So the answer for part 1 is k=1 and C=e^2.

• Erdos
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In this kind of questions values of C does not matter, you just need to show that for some C and K the inequality holds and that's what we have done here.

• Jack Smith
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This is actually a homework question for a course I am already on thin ice on. The grader is kinda sadistic and I am pretty sure he won't give me any points unless I solve the first part(C and K) followed by the second (inequality).

• Jack Smith
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Please understand my plight and sorry I am coming across as annoying.

• Erdos
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Sir, the answer for part 1 i: s k=1 and C=e^2. The answer for part 2 is in the file I uploaded. You seem to be overthinking this. Everything is ok, relax and submit your homework.