Let `f: R->R` be the signum function defined as `f(x)={1,\ x >0,\ \ \ \ 0,\ x=0,\ \ \ \ \-1,\ x<0` and `g: R->R` be the greatest integer function given by `g(x)=[x]` . Then, prove that `fog` and `gof` coincide in `(0,\ 1]` .

Explanation

We have given two functions `f:A->B `and `g:B->C`, then composition of f and g,` gof:A->C` by

`gof(x)=g(f(x)) `for all`x in A `

As a signum function `f:R->R`defined by

`f(x)={1,x`>`0`

`0,x=0`

`−1,x`<`0}`

and the greatest integer function `g:R->R` defined by

`g(x)=[x]` greatest integer less than or equal to x,where `x in (0,1]`

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