Thread: Determining the order of a function with big Oh

1. Determining the order of a function with big Oh

I was wondering if anyone could double check my answers below please.

Give the order of the following functions,

$\displaystyle 1. Ta(n) = 20^2 + (n + 4)^3$
$\displaystyle 2. Tb(n) = (6n + 4)^2 + 3nlog2(n)$
$\displaystyle 3. Tc(n) = (7n + 1)^2log10(n)$

I got the following orders:-

$\displaystyle 1. \theta(n^2)$
$\displaystyle 2. \theta(n log n)$
$\displaystyle 3. \theta(log n)$

Item 3 is the most effecient for very large values of n.

2. 1) the expression of the third order since it contains $\displaystyle n^3$ which dominates the whole expression, hence you get $\displaystyle \theta(n^3)$
2) you have a part which is of the second order: $\displaystyle n^2$ and another one of order $\displaystyle nlog_{2}(n)$ which is lower than $\displaystyle n^2$ hence the total order is $\displaystyle \theta(n^2)$
3) Here, you have multiplication of a polynomial of the second order with $\displaystyle log_{10}(n)$ hence the order is $\displaystyle \theta(n^2log_{10}(n))$

Clearly, the most efficient is the second one since $\displaystyle n^2 < n^2log_{10}(n) < n^3$