Answer
$(-\infty,1]$ and any interval included in it
$[1,\infty)$ and any interval included in it
Work Step by Step
We are given the function:
$f(x)=x^2-2x+8=(x^2-2x+1)-1+8=(x-1)^2+7$
We graph the function.
The vertex is $(1,7)$.
The domain of $f$ is $(-\infty,\infty)$.
The range of $f$ is $[7,\infty)$.
Using the Horizontal Line Test we notice that any horizontal line in the range of the function intersects the graph in two points. Therefore the function $f$ doesn't have an inverse on $(-\infty,\infty)$.
We determine a restriction of the function's domain on which the function has an inverse:
$(-\infty,1]$ and any interval included in it
$[1,\infty)$ and any interval included in it
