1. Let $f(x)=|x-1|+|2 x-1|+|3 x-1|$, where $x \in \mathbf{R}$. Then the range of $f(x)$ is $\qquad$
$-1 .[1,+\infty)$.
Notice,
$$
f(x)=|x-1|+2\left|x-\frac{1}{2}\right|+3\left|x-\frac{1}{3}\right| \text {. }
$$

When $\frac{1}{3} \leqslant x \leqslant \frac{1}{2}$, $f(x)$ reaches its minimum value.
We can take $x=\frac{1}{3}, f_{\text {min }}=f\left(\frac{1}{3}\right)=1$.