# Q. 18 Let $f : R \rightarrow R$ be the Signum Function defined as        $f(x) = \left\{\begin{matrix} 1 & x> 0 \\ 0 &x = 0 \\ -1& x < 0 \end{matrix}\right.$and $g : R \rightarrow R$ be the Greatest Integer Function given by $g (x) = [x]$, where $[x]$ is greatest integer less than or equal to $x$. Then, does $fog$ and $gof$ coincide in $(0, 1]$?

$f : R \rightarrow R$ is defined as $f(x) = \left\{\begin{matrix} 1 & x> 0 \\ 0 &x = 0 \\ -1& x < 0 \end{matrix}\right.$

$g : R \rightarrow R$  is defined as $g (x) = [x]$

Let  $x \in (0, 1]$

Then we have , $[1]=1$ if x=1   and $[x]=0\, \, \, \, \, if\, 0< x< 1$

$\therefore \, \, \, \, fog(x)=f(g(x))=f(\left [ x \right ]) = \left\{\begin{matrix}f(1) & x= 0 \ \\ f(0)& x \in (0,1) \end{matrix}\right = \left\{\begin{matrix}1 & x= 0 \ \\ 0& x \in (0,1) \end{matrix}\right$

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

Hence,for $x \in (0, 1]$ ,  $fog(x)=0$   and $gof(x)=1$.

Hence , gof and fog do not coincide with $(0, 1]$.

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