Problem $3$

Considering function $f(n)0$ and constant $a>0,$ indicate whether each of the following statements

is true. Prove the statements that are true by providing a formal argument that is based on the

definition of asymptotic notation. For statements that are false, provide a counter$-$example to prove

that they are false.

$($a$)f(n)+a=O(f(n)).$

$($b$)$ If $f(n)1,$ then $f(n)+a=O(f(n)).$

$($c$)f(2n)=(f(n)).$

$($d$)$ If $f(n)=O(n^a),$ then $f(2n)=O(n^a).$

$($e$)$ If $f(n)=(n^a),$ then $f(2n)=(f(n)).$