How to Write a Piecewise Function in LaTeX

Why it happens

cases is an amsmath environment: a two-column array with a brace on the left, values in the first column and conditions in the second, separated by &. Without amsmath, \begin{cases} falls back to the older plain TeX \cases macro, which expects a different syntax. So the first symptom of a missing \usepackage{amsmath} is "Misplaced alignment tab character &", not a missing-environment error.

Example

\usepackage{amsmath}
% in the document:
\[
f(x) =
\begin{cases}
  x^2 & \text{if } x \ge 0 \\
  -x  & \text{if } x < 0
\end{cases}
\]

Conditions, commas and "otherwise"

Everything inside cases is math, so a word typed directly comes out as italic variables with its spaces removed: "if x" becomes "ifx". Put words in \text{...} and keep the space inside the braces, as in \text{if } x > 0. \text{otherwise} is the usual last row.

Style guides differ on whether the value ends with a comma and whether the condition starts with "if". Pick one form and use it throughout. For a wider gap between value and condition, add \quad after the &.

\usepackage{amsmath}
% in the document:
\[
|x| =
\begin{cases}
  x,  & \quad x \ge 0, \\
  -x, & \quad \text{otherwise}.
\end{cases}
\]

Full-size fractions: dcases from mathtools

cases sets its rows in text style, so a \frac inside it is as small as in inline math. The mathtools package (which loads amsmath) adds dcases, which sets the rows in display style.

dcases* goes one step further: its second column is ordinary text, so conditions need no \text{...}, and any math in them goes in $...$. mathtools also adds rcases, a brace on the right that groups several lines into one conclusion.

\usepackage{mathtools}
% in the document:
\[
g(x) =
\begin{dcases}
  \frac{1}{x} & x \neq 0 \\
  0           & \text{otherwise}
\end{dcases}
\qquad
h(x) =
\begin{dcases*}
  \frac{\sin x}{x} & when $x \neq 0$ \\
  1                & otherwise
\end{dcases*}
\]

Numbering a piecewise function

cases is not numbered on its own. Inside \begin{equation} ... \end{equation} the whole function gets one number, which you can \label and cite with \eqref.

To number each row, load the cases package and use numcases. Its argument is the left-hand side, and its second column is in text mode, so conditions are plain words with any math in $...$. Each row can carry its own \label.

cases also works inside align when the function is one step of a longer derivation. Put the alignment & before \begin{cases}; the & inside belong to cases.

\usepackage{amsmath}
\usepackage{cases}
% in the document:
\begin{numcases}{f(x)=}
  x^2 & if $x \ge 0$ \label{eq:pos} \\
  -x  & if $x < 0$  \label{eq:neg}
\end{numcases}
For $x \ge 0$, see \eqref{eq:pos}.

Errors and what they mean

"Misplaced alignment tab character &": amsmath is not loaded, or the & sits outside any alignment environment. Add \usepackage{amsmath}.

"Missing $ inserted": cases was used outside math mode. Wrap it in \[ ... \], an equation environment, or $...$ for inline use.

"Extra alignment tab has been changed to \cr": a row has more than one &. cases has exactly two columns; for more, use an array inside \left\{ ... \right.

Conditions in italics with the spaces gone: the words are not inside \text{...}.