How to write conditional equations with one-sided curly brackets

I think this is a job for cases from the amsmath package

\documentclass{article}
\usepackage{amsmath}

\begin{document}
\[
    f(x)= 
\begin{cases}
    \frac{x^2-x}{x},& \text{if } x\geq 1\\
    0,              & \text{otherwise}
\end{cases}
\]
\end{document}

or if you would prefer a displaystyle fraction, then you could use dcases from the mathtools package, which extends (and loads) the amsmath package.

\documentclass{article}
\usepackage{mathtools}

\begin{document}
\[
    f(x)= 
\begin{dcases}
    \frac{x^2-x}{x},& \text{if } x\geq 1\\
    0,              & \text{otherwise}
\end{dcases}
\]
\end{document}

Just for the sake of completeness: There is a pseudo-parenthesis . that can be used to terminate an opening parenthesis:

\documentclass{standalone}

\begin{document}
$\left\{a+b=c\right.$
\end{document}

Output: