Clean-Looking Piecewise-Defined Functions

A simpler solution for aligning fractions is to let TeX decide what space to add:

\[
 f(n) =
  \begin{cases} 
      \hfill \frac{n}{2}    \hfill & \text{ if $n$ is even} \\
      \hfill -\frac{n-1}{2} \hfill & \text{ if $n$ is odd} \\
  \end{cases}
\]

As explained in the comments, \hfill takes up as much space as possible, so putting one on both sides will have the effect of centering the text. If you do this in both lines, you don't have to figure out which one is longer. This is more resistant to changes in your code than using \phantom. I have also gone ahead and changed \textrm to \text, since this what \text is intended for.

If you question is "how do I align my fractions"?, then boom:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
A function is defined:
\[
f(X) = 
\begin{cases}
  \phantom{-}x & \text{if}\ x>0 \\
  -x           & \text{if}\ x<0
\end{cases}
\]
\end{document}

\phantom adds whitespace the size of its argument.

Also, with suitable linebreaking and spacing (which LaTeX ignores) you can make it fairly readable (which seems to answer your other question). emacs has align-current which aligns LaTeX tables at their &s which is super cool.

Here is just an informative, long comment.

amsmath defines the cases environment as

\left\lbrace
\def\arraystretch{1.2}%
\array{@{}l@{\quad}l@{}}%
% cases content
\endarray\right.%

As such, achieving the same spacing as cases with some flexibility in terms of horizontal alignment of the elements contained within, you could use

\documentclass{article}
\usepackage{amsmath}% http://ctan.org/pkg/amsmath
\usepackage{amsfonts}% http://ctan.org/pkg/amsfonts
\begin{document}
For every $n \in \mathbb{N}$, let
\[
  f(n) = \left\{\def\arraystretch{1.2}%
  \begin{array}{@{}c@{\quad}l@{}}
    \frac{n}{2} & \text{if $n$ is even}\\
    -\frac{n-1}{2} & \text{if $n$ is odd}\\
  \end{array}\right.
\]
\end{document}