Find the range of (x^2+x+1) / (x^2-x+1).

Question
Find the range of (x^2+x+1) / (x^2-x+1).

Answer
We can write the given function as:
1 + 2x/(x^2-x+1)
Now rewrite this as
1 + 2/(x+(1/x)-1).

Now, we know the range of x+1/x which is (-INF,-2] U [2,INF).
So, x+1/x >= 2 implies
x+1/x-1 >= 1 implies
1/(x+(1/x)-1) <= 1 implies
1 + 2/(x+(1/x)-1) <= 3 .
Similarly, x+1/x <= -2 implies
1 + 2/(x+(1/x)-1) >= 1/3.

Hence range of the given function is [1/3,3].