Let f: R rightarrow R be the Signum Function | vedclass

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(N/A) It is given that $f: R \rightarrow R$ is the Signum function and $g: R \rightarrow R$ is the Greatest Integer function $g(x) = [x]$.For any $x \

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(N/A) It is given that $f: R ightarrow R$ is the Signum function and $g: R ightarrow R$ is the Greatest Integer function $g(x) = [x]$.For any $x \in (0, 1]$,we analyze the compositions $fog(x)$ and $gof(x)$.First,consider $fog(x) = f(g(x)) = f([x])$.If $x = 1$,then $g(1) = [1] = 1$,so $f(g(1)) = f(1) = 1$.If $x \in (0, 1)$,then $g(x) = [x] = 0$,so $f(g(x)) = f(0) = 0$.Thus,$fog(x) = \begin{cases} 1, & x = 1 \ 0, & x \in (0, 1) \end{cases}$.Next,consider $gof(x) = g(f(x))$.Since $x \in (0, 1]$,$x > 0$,therefore $f(x) = 1$ for all $x \in (0, 1]$.Thus,$gof(x) = g(1) = [1] = 1$ for all $x \in (0, 1]$.Comparing the two,for $x \in (0, 1)$,$fog(x) = 0$ while $gof(x) = 1$.Therefore,$fog$ and $gof$ do not coincide in $(0, 1]$.

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