f(x) = x²-x+1/x²+x+1 find domain and range
explain too
thanks !

Dear Student,

y = $\frac{x^2-x+1}{x^2+x+1}$
Domain is x $\in$R
For Range of function:

yx^​2 + xy + y = x² - x + 1
(y-1)x² + (y+1)x + y-1 = 0

if y = -1
-2x² -1 -1 = 0
-2x² -2 = 0
Therefore, y cannot be equal to -1 since x will be a complex value

Now, y not equal to -1. 
(y-1)x² + (y+1)x + y-1 = 0   has real roots

hence, (y+1)² - 4 (y-1)(y-1) $\ge$0
y² + 2y + 1 - 4y² + 8y - 4 $\ge$0
-3y² + 10y - 3 $\ge$ 0
3y² - 10y + 3 $\le$0
3y² - 9y - y + 3 $\le$ 0
3y (y-3) - 1 (y-3) $\le$0
(3y-1)(y-3) $\le$ 0
y $\in$$\left[\frac{1}{3},3\right]$ 
Hence, range of function is $\left[\frac{1}{3},3\right]$
Regards