Big$-$O notation. We have learnt big$-$O notation to compare the growth rates of func$-$

tions, this exercise helps you to better understand its definition and properties.

$($a$)(12$ points$)$ Suppose n is the input size, we have the following commonly seen functions in complexity analysis: f$1($n$)=1,$ f$2($n$)=$ log n$,$ f$3($n$)=$ n$,$ f$4($n$)=$ n log n$,$ f$5($n$)=$ n$2,$ f$6($n$)=2$n$,$ f$7($n$)=$ n$!.$ Intuitively, the growth rate of the func$-$ tionssatisfy$1$