Question

Let $f:R\to R$ be the Signum function defined as $f\left(x\right)=\{\begin{array}{cc}1,& x>0\\ 0,& x=0\\ -1,& x<0\end{array}$ and $g:R\to R$ be the

greatest integer function given by g(x) =[x] is greatest integer less than or equal to x. Then, fog and gof coincide in (0,1].

Answer 1

It is given that $f:R\to R$ is defined $f\left(x\right)=\{\begin{array}{cc}1,& x>0\\ 0,& x=0\\ -1,& x<0\end{array}$

Also, $g:R\to R$ is defined as g(x)=[x], where [x] is the greatest integer less than or equal to x. Now, let $x\in (0,1]$. Then, we have
[x]=1 if x =1 and [x]=0 if 0 < x < 1
$\therefore fog\left(x\right)=f\left(g\right(x\left)\right)=f\left(\right[x\left]\right)=\{\begin{array}{c}f\left(1\right),ifx=1\\ f\left(0\right),ifx\in (0,1)\end{array}=\{\begin{array}{c}1,ifx=1\\ 0,x\in (0,1)\end{array}$
gof(x)=g(f(x))=g(1)=[1]=$1[\because x>0]$
Then, when $x\in (0,1)$, we have fog(x)=0 and gof(x)=1.
But fog(1)$\ne$ gof(1)
Hence, fog and gof do not coincide in (0,1].