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

Explanation

Let `y=(x^(2)-x+1)/(x^(2)+x+1)`
or `(1-y)x^(2)-(1+y)x+1-y=0`
Now, x is real. Then
`D ge 0`
or `(1+y)^(2)-4(1-y)^(2) ge 0`
or `(1+y-2+2y)(1+y+2-2y) ge 0`
or `(3y-1)(3-y) ge 0`
or `3(y-(1)/(3))(y-3) le 0`
or ` (1)/(3) le y le 3`
Hence, the range is `[(1)/(3),3]`.