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Consider the following function $f(x)$ = $x^8$+6$x^7$-9$x^5$-$x^4$+2$x^2$-18. Which of the following is true if x is greater than 56?

  1. $f(x)$ = O($x^8$)
  2. $f(x)$ = Ω($x^8$)
  3. $f(x)$ = θ($x^8$)
  4. $f(x)$ = None of the above.
in Algorithms by Active (4.8k points)
edited by | 127 views


we ignore the constants and lower order term. 

what's the answer given?

it should be A
Should be C as for large enough vale of $x$, $f(x)$ has to be greater than $x^8$ (taking constant as 1 for lower bound).
I too think the answer should be A, obviously for the same reason as stated by Utkarsh

But C is provided as answer.

 if C is the answer then you cannot say A is wrong 

Yes, A isn't wrong. Both A and C are correct - however I assumed that we should give as tight a bound as possible and hence said C is correct.
ok i agree with you $C$ is more appropriate
A) will be answer

because it can be 57 as lower bound and upperbound $n^{8}$

@goxul Please explain me your approach. How C is more appropriate.

1 Answer

+1 vote
C is correct because we can find two constants $c_1$ and $c_2$ such that: $c_1 x^8 \leq f(x) \leq c_2 x^8$.

For the LHS to be true, put $c_1 = 1$, RHS will be true for a large value of $c_2$.

Thus we can say that $f(x) \in \Omega(x^8)$
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