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.`

Explanation

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`