Let $f : R \to R$ be the Signum function defined as
$⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 1, & x > 0 0, & x = 0 - 1, & x < 0 \end{matrix}$
and $g : R \to R$ be the Greatest Integer Function given by $g (x) = [x]$ , where $[x]$ is greatest integer less than or equal to $x$ . Then, does $f o g$ and $g o f$ coincide in $(0, 1]$ ?

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Solution

Given two functions $f : A \to B$ and $g : B \to C$ , then composition of $f$ and $g$ , $g o f : A \to C$ by
$g o f (x) = g (f (x)) \forall x ϵ A$
Since a signum function $f : R \to R$ defined by
$f (x) = {1, x > 0 0, x = 0 - 1, x < 0}$
and the greatest integer function $g : R \to R$ defined by
$g (x) = [x]$ greatest integer less than or equal to $x$ ,where $x ϵ (0, 1]$
Step 1:Calculating $g o f$
$x ϵ (0, 1] \Rightarrow f (x) = 1 a s x > 0 ∴ g o f = g (f (x)) = g (1) = [1] = 1$
Step 2:Calculating $f o g$
$x ϵ (0, 1] \Rightarrow g (x) = [1] = 1 i f x = 0 o r g (x) = [0]$
$i f x ϵ (0, 1)$
$\Rightarrow f o g = f (g (x)) = {f (1) x = 1 f (0) x ϵ (0, 1)}$
$= {1 x = 1 0 x ϵ (0, 1)}$
Thus $x ϵ (0, 1] f o g (x) = 0$ and $g o f (x) = 1$ .
$∴$ they do not coincide.

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