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Question

Let f:RR be the Signum function defined as
1,x>00,x=01,x<0
and g:RR 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]?

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Solution

Given two functions f:AB and g:BC, then composition of f and g, gof:AC by
gof(x)=g(f(x))xϵA
Since a signum function f:RR defined by
f(x)={1,x>00,x=01,x<0}
and the greatest integer function g:RR defined by
g(x)=[x] greatest integer less than or equal to x,where xϵ(0,1]
Step 1:Calculating gof
xϵ(0,1]f(x)=1asx>0gof=g(f(x))=g(1)=[1]=1
Step 2:Calculating fog
xϵ(0,1]g(x)=[1]=1ifx=0org(x)=[0]
ifxϵ(0,1)
fog=f(g(x))={f(1)x=1f(0)xϵ(0,1)}
={1x=10xϵ(0,1)}
Thus xϵ(0,1]fog(x)=0 and gof(x)=1.
they do not coincide.

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