Suppose $f$:$xx$ is a function with the same domain and codomain. For each a positive integer $n,$ let $f^n$:$xx$ be the function with the formula $f^n(x)=$ubrace$(f(f(f(cdotsfubrace{)}_{n\text{t}\text{i}\text{m}\text{e}\text{s}}(x)cdots)))$ so that $f^1(x)=f(x),f^2(x)=f(f(x)),$ and $f^3(x)=f(f(f(x))),$ and so on$.$ The function $f$ has finite order if there is a positive integer $n$ with $f^n=i{d}_x.$ When $f$ has finite order, its order is the smallest positive integer $n$ with $f^n=i{d}_x.$ How many functions $f$:$\{1,2,$dots,$n\}\{1,2,$dots,$n\}$ have finite order? What are the possible values for their orders? Justify your answer.