Extend inverses, and slightly clarify. (modelica#2478)

* Extend inverses, and slightly clarify.

chapters/functions.tex

@@ -1616,12 +1616,13 @@ \section{Declaring Inverses of Functions}\doublelabel{declaring-inverses-of-func
 parameters with and without default value. The restriction is that the
 \emph{number of unknown variables} (see \autoref{balanced-models}) in the output formal
 parameter of both $f_1$ and $f_2$ must be
-the same and that $f_2$ must have exactly the same formal
-parameters as $f_1$ (with the same defaults, if a formal
-parameter $u_m$ has a default), but the order of the formal
+the same and that $f_2$ should have a union of output and formal
+parameters that is the same or a sub-set of that union for $f_1$, but the order of the formal
 parameters may be permuted.
 
+
 \begin{example}
+Same union of variables:
 \begin{lstlisting}[language=modelica]
 function h_pTX
   input Real p "pressure";
@@ -1644,6 +1645,31 @@ \section{Declaring Inverses of Functions}\doublelabel{declaring-inverses-of-func
 \end{lstlisting}
 \end{example}
 
+The sub-set case is useful if $f_1$ computes the inverse of $f_2$ within a region, or up to a certain tolerance.
+Then, $f_1$ may specify $f_2$ as inverse with fewer arguments, skipping the arguments for tolerance and/or the region.
+
+\begin{example}
+
+\begin{lstlisting}[language=modelica]
+function inv_sine
+  input Real x;
+  input Real angleOrig;
+  output Real angle;
+  // Finds sine(angle)=x with angle closest to angleOrig.
+algorithm
+  ...
+  annotation(inverse(x=sine(angle)));
+end inv_sine;
+
+function sine
+  input Real angle;
+  output Real x;
+algorithm
+  x:=sin(angle);
+  // Note: No inverse.
+end sine;
+\end{lstlisting}
+\end{example}
 \section{External Function Interface}\doublelabel{external-function-interface}
 
 Here, the word function is used to refer to an arbitrary external