Homework Help: Prove this inequality

1. May 10, 2013

utkarshakash

1. The problem statement, all variables and given/known data
If f and g are two distinct linear functions defined on R such that they map[-1,1] onto [0,2] and h:R-{-1,0,1}→R defined by h(x)=f(x)/g(x) then show that |h(h(x))+h(h(1/x))|>2

2. Relevant equations

3. The attempt at a solution
I assume f(x) to be ax+b and g(x) to be lx+m so that h(x) is (ax+b)/(lx+m). From here I can write h(h(x)) and h(h(1/x)) but there is nothing I can see that will help me to prove this inequality. Any ideas?

2. May 11, 2013

SammyS

Staff Emeritus
You can be more specific regarding the functions, f & g .

There are only two distinct linear functions which map [-1,1] onto [0,2] .

What are they?

3. May 11, 2013

utkarshakash

y=x+1 and y=-x+1. Are these correct?

4. May 11, 2013

SammyS

Staff Emeritus
Yes. Equivalently, y=1+x and y=1-x .

So, there are only two cases to consider.

Choosing f(x) = 1+x and g(x) = 1-x, for now, what is h(1/x) ?

5. May 11, 2013

utkarshakash

x+1/x-1 which can be reduced to -f(x)/g(x)

And after simplifying further I am left with proving this inequality

|x-(1/x)|>2

Last edited: May 11, 2013
6. May 12, 2013

SammyS

Staff Emeritus
It looks like you get h(1/x) = -f(x)/g(x), for either way of assigning 1-x and 1+x to f(x) and g(x).
I got something similar to x-(1/x), but it is different.