資料結構與演算法

(4 points) For positive functions $f(n)$ and $g(n)$ on $\mathbb{N}$. If $f(n) = O(g(n))$ and $f(n) = \Omega(1)$, how many of the following statements are true?

• $g(n) = \Omega(f(n))$
• $f(n) \cdot \log(1 + f(n)) = O\big(g(n) \cdot \log(1 + g(n))\big)$
• $2^{f(n)} = O\big(2^{g(n)}\big)$
• $(f(n))^k = O\big((g(n))^k\big)$ for any $0 < k < 1$