Gateforum Test Series: Algorithms - Asymptotic Notations

Question

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?

• A. $f(x)$ = O($x^8$)
• B. $f(x)$ = Ω($x^8$)
• C. $f(x)$ = θ($x^8$)
• D. $f(x)$ = None of the above.

Comments

ok i agree with you $C$ is more appropriate

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A) will be answer

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

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

Answer 1

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 $f(x) \in \Omega(x^8)$