Question

Let f : $R\to R$ be the Signum Function defined as
$f\left(x\right)=\{\begin{array}{cc}1,& x>0\\ 0,& x=0\\ -1,& x<0\end{array}\text{and}g:R\to R$
be the Greatest Integer Function given by $g\left(x\right)=x$, where $\left[x\right]$ is greatest integer less than or equal to $x$. Then does $fog$ and gof coincide in $(0,1]$ ?

Answer 1

Given, $f\left(x\right)$ : $R\to R$ be the Signum Function defined as
$f\left(x\right)=\{\begin{array}{cc}1,& x>0\\ 0,& x=0andg:\left(x\right)=\left[x\right]\\ -1,& x<0\end{array}$
Now,
$gof=g\left(f\right(x\left)\right)=\left[f\right(x\left)\right]$
In $(0,1],f\left(x\right)=1$
So, $gof=\left[1\right]=1fog=f\left(g\right(x\left)\right)=f\left(\right[x\left]\right)$
$=\{\begin{array}{cc}1,& \left[x\right]>0\\ 0,& \left[x\right]=0\\ -1& \left[x\right]<0\end{array}$

$fog=f\left(g\right(x\left)\right)=f\left(\right[x\left]\right)=\{\begin{array}{cc}1,& \left[x\right]>0\\ 0,& \left[x\right]=0\\ -1& \left[x\right]<0\end{array}$

$f\left(\right[x\left]\right)=\{\begin{array}{cc}1,& x\ge 1\\ 0,& 0\le x<1\\ -1,& x<0\end{array}$
As, $x\in (0,1]$
$fog=f\left(\right[x\left]\right)=\{\begin{array}{cc}0,& 0<x<1\\ 1,& x=1\end{array}$
$gof=1fog=\{\begin{array}{cc}0,& 0<x<1\\ 1,& x=1\end{array}$
In $gof$, there is only one value, & in $fog$, there are two values.
Hence, $fog$ and $gof$ dop not coincide in $(0,1]$.