Question $4$

Suppose we are given positive functions $f,g,h$ and i such that for all $n1$ we have $f(n)1,g(n)1,h(n)1$ and $i(n)1.$ Further assume that $f(n)$ is $O(g(n)),g(n)$ is $(h(n))$ and $f(n)$ is $(i(n)).$ Justify if the statements below are true, false, or possibly true$/$possibly false.

$($a$)h(n)$ is $O(g(n))$

$($b$)f(n)$ is $(h(n))$

$($c$)g(n)$ is $(i(n))$

$($d$)2f(n)+5i(n)+6h(n)$ is $O(g(n))$

$($e$)z(n)$ is $(y(n))$ where the functions $z(n)$ and $y(n)$ are defined as follows: $z(n)=2^{f(n)}$ and $y(n)=2^{i(n)}$