$(30$ points$)$ For each of the following two functions f$($n$)$ and g$($n$),$ indicate whether f $=$ O$($g$),$ or f $=\backslash$Omega $($g$)$ or both $($in which case f $=\backslash$Theta $($g$)).$

$($a$)$ f$($n$)=$n$100$andg$($n$)=$n$200.($b$)$ f$($n$)=$ n$1/2$ and g$($n$)=$ n$2/3.$

$($c$)$ f$($n$)=100$n$+$lgnandg$($n$)=$n$+($lgn$)2.($d$)$ f$($n$)=$ nlgn and g$($n$)=10$nlg$(10$n$).$

$($e$)$ f$($n$)=10$lgn and g$($n$)=$ lg$($n$2).($f$)$ f$($n$)=$ n$2/$lgn and g$($n$)=$ n$($lgn$)2.$

$($g$)$ f$($n$)=$ n$0\mathrm{.}1$ and g$($n$)=($lg n$)10.($h$)$ f$($n$)=$n and g$($n$)=($lg n$)3.$

$($i$)$ f$($n$)=$ n$2$n and g$($n$)=3$n$.($j$)$ f$($n$)=2$n and g$($n$)=2$n$+1.$