Range of $f(x) = \frac{1}{x^2 - x +1}$
Since$$x^2-x+1=\left(x-\frac12\right)^2+\frac34,$$the range of $x^2-x+1$ is $\left[\frac34,\infty\right)$, and therefore the range of your function is $\left(0,\frac43\right]$.
Since$$x^2-x+1=\left(x-\frac12\right)^2+\frac34,$$the range of $x^2-x+1$ is $\left[\frac34,\infty\right)$, and therefore the range of your function is $\left(0,\frac43\right]$.