For each of the following functions, decide the asymptotic relationships between f$($n$)$ and g$($n$).$ i$.$e$.,$ decide if f$($n$)0($g$($n$)),$ f$($n$)=2($g$($n$)),$ or f$($n$)=($g$($n$)).$ For full marks you must provide proofs that justify your answers. All logarithms are base $2$ unless otherwise stated. $($a$)$ f$($n$)=5$n$28($n$)=$ n$2($b$)$ f$($n$)=3$n$6+4$n$2+58($n$)=$ n$5($c$)$ f$($n$)=$ logn$+1$ n g$($n$)=$ logn logn $($d$)$ f$($n$)=8($n$)=($e$)$ f$($n$)=2"8($n$)=22$n $($f$)$ f$($n$)=2$k logn $=8($n$)=$ nk n$<22022($g$)$ f$($n$)=$ g$($n$)=2202222022,,$ n$>22022($h$)$ f$($n$)=2$Vlogn $8($n$)=($logn$)1008($n$)=$ nlog logn $($i$)$ f$($n$)=($logn$)$logn $($j$)$ f$($n$)=32"8($n$)=2214$