Consistently specify a Function and use it for computation and plotting

Question

I would like to be able to specify a mathematical function and be able to use that for multiple purposes including:

1. Computation of values (see bullet points below)
2. Graphing using pgfplot
3. Graphing using pgfplots/gnuplot

This could be considered a followup to: Consistently specify function to be graphed with or without gnuplot which addresses points 2 and 3 where it was pointed out that with the definition of a function as

\newcommand*{\FunctionF}{(x)^3}%

one could use it to plot with just PGFplots, or in conjunction with gnuplot:

\addplot {\FunctionF};
\addplot gnuplot {\FunctionF};

But, I desire a way to also be able to use the same definition to do computations such as where

• I want to define a piecewise function as in Defining a Piecewise Function for PGFplots,

• be able to use it to compute values of individual points as in this example where I label a specific point given the x value

• be able to use it as any other built in math function and define a translation as in the green curve below

So to produce the blue line graph, I have used the \pgfmathdeclarefunction definition which solves problems 1 and 2, and the above referenced question uses the \newcommand*{\FunctionF}{(x)^3} definition to solve problems 2 and 3.

Question: Is there someway of solving problems 1, 2 and 3 all with one definition of the function?

\documentclass[border=5pt]{standalone}
\usepackage{pgfplots}

\newcommand*{\XAxisMin}{-1.0}
\newcommand*{\XAxisMax}{3.0}
\newcommand*{\YAxisMin}{-2.0}
\newcommand*{\YAxisMax}{10}

\newcommand*{\DomainMinF}{\XAxisMin}
\newcommand*{\DomainMaxF}{2.2}

\pgfkeys{/pgfplots/Axis Style/.style={
    xmin=\XAxisMin, xmax=\XAxisMax,
    ymin=\YAxisMin, ymax=\YAxisMax, 
    domain=\DomainMinF:\DomainMaxF,
    width=6.5cm
}}

% Gnuplot options here  have no effect if not using GnuPlot
\pgfkeys{/pgfplots/Plot Style/.style={
    translate gnuplot=true,% can use ‘^’ instead of ‘**’
    id=foo,
    mark=none,
    domain=\DomainMinF:\DomainMaxF,
    samples=50,
    ultra thick,
}}

\newcommand*{\AddLabel}[1]{\node [align = center] at (axis cs: 0.25,7.1) {#1};}%
\newcommand*{\LabelPoint}[2]{\addplot [mark=*] coordinates {(#1,#2)} node [right] {\footnotesize$(#1,#2)$};}%
%-----------------------------

% I would like to only have to specify the function here once
\pgfmathdeclarefunction{FunctionF}{1}{\pgfmathparse{(#1)^(3)}}%
\newcommand*{\FunctionFGnuplot}{(x)^3}%

% Define translation of F
\pgfmathdeclarefunction{FunctionG}{1}{\pgfmathparse{FunctionF(#1+1.0)}}%


\newcommand*{\SpecialXValue}{1.1}%

\begin{document}
\begin{tikzpicture}
\begin{axis}[Axis Style]
    \addplot [Plot Style, blue] ({x},{FunctionF(x)});
    \addplot [Plot Style, green]({x},{FunctionG(x)});

    \AddLabel{1. without \\ Gnuplot}
    \node at (axis cs: 1.8,3.0) [blue] {$f(x)$};
    \node at (axis cs: -0.2,3.0) [green] {$f(x+1)$};

    \pgfmathsetmacro{\YatSpecialX}{FunctionF(\SpecialXValue)}
    \LabelPoint{\SpecialXValue}{\YatSpecialX}
\end{axis}
\end{tikzpicture}
%
\begin{tikzpicture}
\begin{axis}[Axis Style]
    \addplot [Plot Style, red] gnuplot {\FunctionFGnuplot};
    \AddLabel{2. with Gnuplot \\
            \footnotesize(but resorted to using \\
            \footnotesize pgfmathdeclarefunction \\
            \footnotesize to compute point)}

    % How do I do something like this with \FunctionFGnuplot
    \pgfmathsetmacro{\YatSpecialX}{FunctionF(\SpecialXValue)}
    \LabelPoint{\SpecialXValue}{\YatSpecialX}
\end{axis}
\end{tikzpicture}
\end{document}

Answer 1

If you define your function using \def\<function macro>(#1){\pgfmathparse{...#1...}, you can use the function in all the applications you mentioned by saying \<function macro>(x) or \<function macro>(x-1), or \<function macro>(2.4). This will work both when using Gnuplot and when using straight PGFplots:

\documentclass[border=5pt]{standalone}
\usepackage{pgfplots}

\def\FunctionF(#1){(#1)^3}%

\newcommand*{\SpecialXValue}{1.5}%

\begin{document}
\begin{tikzpicture}
\begin{axis}[
    domain=-3:3, samples=50,
    enlargelimits=false,
    ymin=-5, ymax=5,
    every axis plot post/.style= ultra thick]
    \addplot [red] gnuplot {\FunctionF(x)};%
    \addplot [blue] gnuplot {\FunctionF(x+1)};%
    \addplot [cyan] {(x<1)*(3-3*x)+!(x<1)*-\FunctionF(x-1.5))};
    \pgfmathsetmacro{\specialY}{\FunctionF(-1.5)}
    \fill (axis cs:-1.5,\specialY) circle [radius=2pt] node [right] {(2,\specialY)};
\end{axis}


\end{tikzpicture}
\end{document}

---

Old Answer:

You can redefine the x function to return a constant value (in your case \SpecialXValue) before calculating the y value. The easiest way to do that is to issue

\tikzset{declare function={x=\SpecialXValue;}}

just before you calculate the y value.

You can still add new functions after this, the definition of x as a fixed value doesn't persist.

\documentclass[border=5pt]{standalone}
\usepackage{pgfplots}

\newcommand*{\XAxisMin}{-1.0}
\newcommand*{\XAxisMax}{3.0}
\newcommand*{\YAxisMin}{-2.0}
\newcommand*{\YAxisMax}{10}

\newcommand*{\DomainMinF}{\XAxisMin}
\newcommand*{\DomainMaxF}{2.2}

\pgfkeys{/pgfplots/Axis Style/.style={
    xmin=\XAxisMin, xmax=\XAxisMax,
    ymin=\YAxisMin, ymax=\YAxisMax, 
    domain=\DomainMinF:\DomainMaxF,
    width=6.5cm
}}

% Gnuplot options here  have no effect if not using GnuPlot
\pgfkeys{/pgfplots/Plot Style/.style={
    translate gnuplot=true,% can use ‘^’ instead of ‘**’
    id=foo,
    mark=none,%
    domain=\DomainMinF:\DomainMaxF,%
    samples=50,%
    ultra thick,
}}

\newcommand*{\LabelPoint}[2]{\fill (axis cs:#1,#2) circle [radius=3pt] node [right] {\footnotesize$(#1,#2)$};}%


\newcommand*{\FunctionFGnuplot}{(x)^3}%

\newcommand*{\SpecialXValue}{1.1}%

\begin{document}
\begin{tikzpicture}
\begin{axis}[Axis Style]
    \addplot [Plot Style, red] gnuplot {\FunctionFGnuplot};%

    \tikzset{declare function={x=\SpecialXValue;}}
    \pgfmathsetmacro{\YatSpecialX}{\FunctionFGnuplot}%
    \LabelPoint{\SpecialXValue}{\YatSpecialX}
\end{axis}


\end{tikzpicture}
\end{document}