Question

Let `f: R->R`be the Signum Function defined as `f(x)={1,x >0 0,x=0-1,x<1` and `g: R-> R`be the Greatest Integer Function given by `g(x) = [x]`, where [x] is greatest integer less than or equal to x. Then does fo

Answer 1

In f: `R rarr R` ,
`f(x)={(1"," x gt 0 ),(0","x=0),(-1 ","x lt 0 ):}`
and in g : `R rarr R` f(x) = [x]
Let ` x in [0,1] then`
[ x] `={(1 if x=1),(0 if 0 lt x lt 1 ):}`
`therefore (fog) (x) = f(g(x)}=f{[x]}`
`{(f(1)","if x=1 ),(f(0)"," if 0 lt x lt1):}`
`{(1"," if x=1),(0"," if 0 lt x lt 1):}`
and (gof) (x) = g[f(x)]=g(1)=[1]=1
Therefore (fog ) (1) `ne ` (gof)(1)
`rArr ` gof and fog ,[0,1] do not coincide