Mathematics - Functions Question with Solution | TestHub

$g:N\to N$

$g(3n+1)=3n+2$

$g(3n+2)=3n+3$

$g(3n+3)=3n+1$

$g\left(x\right)=\left\{\begin{array}{c}\begin{array}{ll}x+1;& x=3k+1\\ x+1;& x=3k+2\\ x-2;& x=3k+3\end{array}\end{array}\right.\begin{array}{}\\ \\ \end{array}$

$g\left(g\left(x\right)\right)=\left\{\begin{array}{c}\begin{array}{ll}x+2;& x=3k+1\\ x-1;& x=3k+2\\ x-1;& x=3k+3\end{array}\end{array}\right.\begin{array}{}\\ \\ \end{array}$

$g\left(g\left(g\left(x\right)\right)\right)=\left\{\begin{array}{ll}x;& x=3k+1\\ x;& x=3k+2\\ x;& x=3k+3\end{array}\right.\begin{array}{}\\ \\ \end{array}\begin{array}{}\\ \\ \end{array}$

If $f:N\to N,f$ is a one-one function such that $f\left(g\right(x\left)\right)=f\left(x\right)\Rightarrow g\left(x\right)=x$, which is not the case

If $f:N\to N, f$ is an onto function such that $f\left(g\right(x\left)\right)=f\left(x\right)$ one possibility is

$f\left(x\right)=\left\{\begin{array}{ll}n;& x=3n+1\\ n;& x=3n+2\\ n;& x=3n+3\end{array}\right.; n\in N$

Here $f\left(x\right)$ is onto, also $f\left(g\right(x\left)\right)=f\left(x\right)\forall x\in N$