We have  $f\left(x\right)=|x-1|$

Here,$f\left(x\right)$ is a modulus function and since modulus of a real number is uniquely defined $\mathrm{\forall }$ real positive number.

$\therefore$  The domain of $f\left(x\right)\text{ is }R$.

We see that $f\left(x\right)=|x-1|$

$\begin{array}{l}f\left(x\right)=\left\{\begin{array}{cc}x-1,& \text{ if }x\ge 1\\ -(x-1),& \text{ if }x<1\end{array}\right.\\ \Rightarrow f\left(x\right)=\left\{\begin{array}{lll}x-1,& \text{ if }& x\ge 1\\ 1-x,& \text{ if }& x<1\end{array}\right.\end{array}$

From above, we observe that in both cases $f\left(x\right)\ge 0$.

Hence, range of $f\left(x\right)\text{ is }[0,\mathrm{\infty })$.