The range of the function (f(x)= x^2-x+1 x^2+x+1 ) where ( x R ), is

Step-by-step solution

( aligned & Let y= x^2-x+1 x^2+x+1 & x ^2( y -1)+ x ( y +1)+( y -1)=0 & x= -(y+1) (y+1)^2-4(y-1)^2 2(y-1) & = -(y+1) -3 y^2+10 y-3 2(y-1) is real iff & y -1 0 y 1 & If y=1 then original equation gives x =0 , & so taking y =1 & Also 3 y^2-10 y+3 0 & (3 y-1)(y-3) 0 & y [ 1 3 , 3 ] Range is [ 1 3 , 3 ] & aligned )