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2. (3 marks) Let $f(n)$ and $g(n)$ be positive functions such that $f(n)$ is $O(g(n))$ and $g(n) \ge 1$ for all $n \ge 1$. Using the definition of \"big Oh\" show that $f(n) + k$ is $O(g(n))$, where $k > 0$ is constant. Hints. Since $f(n)$ is $O(g(n))$ then there are constants $c > 0$ and integer $n_0 \ge 1$ such that $f(n) \le c'g(n)$ for all $n \ge n_0$. Hence, $f(n) + k \le c'g(n) + k$... Also, recall that $g(n) \ge 1$ for all $n \ge 1$.

          2. (3 marks) Let $f(n)$ and $g(n)$ be positive functions such that $f(n)$ is $O(g(n))$ and $g(n) \ge 1$ for all $n \ge 1$. Using the definition of \"big Oh\" show that $f(n) + k$ is $O(g(n))$, where $k > 0$ is constant.
Hints. Since $f(n)$ is $O(g(n))$ then there are constants $c > 0$ and integer $n_0 \ge 1$ such that $f(n) \le c'g(n)$ for all $n \ge n_0$. Hence, $f(n) + k \le c'g(n) + k$... Also, recall that $g(n) \ge 1$ for all $n \ge 1$.

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2. (3 marks) Let f(n) and g(n) be positive functions such that f(n) is O(g(n)) and g(n) ≥ 1 for all n ≥ 1. Using the definition of b̈ig Ohs̈how that f(n) + k is O(g(n)), where k > 0 is constant.
Hints. Since f(n) is O(g(n)) then there are constants c > 0 and integer n0 ≥ 1 such that f(n) ≤ c'g(n) for all n ≥ n0. Hence, f(n) + k ≤ c'g(n) + k... Also, recall that g(n) ≥ 1 for all n ≥ 1.

Added by Andrew F.

Computer Science and Information Technology

Trishna Knowledge Systems 2018 Edition

Instant Answer

Solved by Expert Yujie Wang

05/20/2022

Step 1

Step 1: Since f(n) is O(g(n)), there are constants c' > 0 and integer n' > 1 such that f(n) <= c'g(n) for all n > n'. Show more…

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2. (3 marks) Let f(n) and g(n) be positive functions such that f(n) is O(g(n)) and g(n) > 1 for all n > 1. Using the definition of "big Oh" show that f(n) + k is O(g(n)), where k > 0 is constant. Hints. Since f(n) is O(g(n)) then there are constants c' > 0 and integer n' > 1 such that f(n) c'g(n) for all n > no. Hence, f(n)+ k c'... Also, recall that g(n) > 1 for all n 1.

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