How to write absolute value as a "true" function Here's the basic absolute value ... what?

\begin{align*}
 |x| = \left\{ \begin{array}{r@{\quad \mathrm{if} \quad}l}
 x  &amp;  x &gt; 0, \\
 0  &amp;  x = 0, \\
 \!\! -x  &amp;  x &lt; 0. \end{array} \right.
 \end{align*}

Is this a function, an "expression," or just some short-hand statement? How would you categorize this? Could it be a function if I wrote it like this?

\begin{align*}
 f(x) = \left\{ \begin{array}{r@{\quad \mathrm{if} \quad}l}
 x  &amp;  x &gt; 0, \\
 0  &amp;  x = 0, \\
 \!\! -x  &amp;  x &lt; 0. \end{array} \right.
 \end{align*}

or is this not really a proper algebra function format?
That is correct; it is a piecewise function. One comment: you may simply define $f(x) = x$ when $x \geq 0$ and $f(x) = -x$ when $x &lt; 0$. There is no need to break it into three pieces by separating out the $x = 0$ case.
The second equation is valid. $f (x) $ is a shorthand notation for some function that takes in some value $x $ on a specified domain and returns some value on a range. So, given the definition of $|x|$ in your question, if you label it as $f (x) $ then you have $f (x)=|x|$ which is essentially what you wrote for your second equation.