The range of function f(x) = x2 - 2x + 2; x ∈ R is:

Given

$f(x)=x^2-2x+2$

Complete the square

$f(x)=x^2-2x+1+1$

$f(x)=(x-1)^2+1$

Since $(x-1)^2 \ge 0$ for all $x \in \mathbb{R}$

Minimum value occurs when

$(x-1)^2=0$

$x=1$

$f(1)=1$

Thus

$f(x)\ge1$

The range is $[1,\infty)$.