If `fa n dg`are two distinct linear functions defined on `R`such that they map `{-1,1]`onto `[0,2]`and `h : R-{-1,0,1}vecR`defined by `h(x)=(f(x))/(g(x)),`then show that `|h(h(x))+h(h(1/x))|> 2.`

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Let two linear functions be

`f(x)=ax +b” and ” g(x) =cx +d.`

They map `[-1,1] to [0,2]` and mapping is onto.

Therefore, `f(-1)=0 and f(1)=2`

`and g(-1)=2 and g(1)=0,` i.e.,

`-a+b=0 and a+b=2 ” (1) ” `

`and -c+d=2 and c+d=0 ” (2) ” `

`or a=b=1 and c= -1, d=1`

`or f(x)=x+1 and g(x)=-x+1`

`or h(x)=(x+1)/(1-x) or h(h(x)) =((x+1)/(1-x)+1)/((x+1)/(1-x)-1)=(1)/(x)`

` or h(h(1//x))=x`

`or |h(h(x))+h(h(1//x))|=|x+1//x| gt 2`