Question

Let $f:R\to R$ be the Signum function defined as
$\{\begin{array}{cc}1,& x>0\\ 0,& x=0\\ -1,& x<0\end{array}$
and $g:R\to R$ be the Greatest Integer Function given by $g\left(x\right)=\left[x\right]$, 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 two functions $f:A\to B$ and $g:B\to C$, then composition of $f$ and $g$, $gof:A\to C$ by

$gof\left(x\right)=g\left(f\left(x\right)\right)\forall xϵA$

Since a signum function $f:R\to R$ defined by

$f\left(x\right)=\{1,x>00,x=0-1,x<0\}$

and the greatest integer function $g:R\to R$ defined by

$g\left(x\right)=\left[x\right]$ greatest integer less than or equal to $x$,where $xϵ(0,1]$

Step 1:Calculating $gof$

$xϵ(0,1]\Rightarrow f\left(x\right)=1asx>0\therefore gof=g\left(f\left(x\right)\right)=g\left(1\right)=\left[1\right]=1$

Step 2:Calculating $fog$

$xϵ(0,1]\Rightarrow g\left(x\right)=\left[1\right]=1ifx=0org\left(x\right)=\left[0\right]$

$ifxϵ(0,1)$

$\Rightarrow fog=f\left(g\left(x\right)\right)=\left\{f\right(1)x=1f\left(0\right)xϵ(0,1)\}$

$=\{1x=10xϵ(0,1)\}$

Thus $xϵ(0,1]fog\left(x\right)=0$ and $gof\left(x\right)=1$.

$\therefore$ they do not coincide.