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]?

Answers (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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