Filling intersections of polynomials and other shapes

Question

I'm trying to re-create a diagram from a textbook, which has various regions defined by polynomials. Ideally I'd like to fill them with the correct colours, to match the text that goes with it.

While I can create and fill the five 'shapes', I'm so far unable to fill the remainder of the background, without also re-filling the other regions (with the exception of the central circle).

Is there an (easy) way to do this? Or do I need to re-think the diagram entirely?

\documentclass[tikz,border=12pt,svgnames]{standalone}

\begin{document}
\begin{tikzpicture}[line cap=round,line join=round,x=0.5cm,y=0.5cm,thin,samples=100,fill opacity=0.5]
\draw[help lines,step=0.5cm] (0,0) grid (14,8);
\draw[fill=red,domain=0:5] plot(\x,{-(1/3)*(\x+1)*(\x-5)}) |- (5,0) -- (0,0) -- cycle;
\draw[fill=Green,domain=10:14] plot(\x,{((\x-12)^3)/4+2}) |- (14,0) -- (10,0) -- cycle;
\draw[fill=Blue,domain=0:4] plot(\x,{((\x-2)^3)/4+6}) |- (4,8) -- (0,8) -- cycle;
\draw[fill=black,domain=9:14] plot(\x,{(\x-9)*(\x-15)/3+8}) |- (14,8) -- (9,8) -- cycle;
\draw[fill=Green,domain=9:14] (7,4) circle (3);
\draw[fill=pink,even odd rule,domain=9:14] (0,0) rectangle (14,8) (7,4) circle (3);
\end{tikzpicture}
\end{document}

Answer 1

You can build the plot/background layers as opaque, then make them transparent as a group over the gridlines.

Result

Code

\documentclass[tikz,border=12pt,svgnames]{article}

\usepackage{tikz}
\newsavebox{\myimagebox}

\begin{document}


% Save top layer in a box
% Composed as opaque layer, then made transparent as a group
\savebox{\myimagebox}{%
\begin{tikzpicture}[line cap=round,line join=round,x=0.5cm,y=0.5cm,thin,samples=100,transparency group,opacity=.5]
\draw[fill=pink,even odd rule,domain=9:14] (0,0) rectangle (14,8) (7,4) circle (3);
\draw[fill=red,domain=0:5] plot(\x,{-(1/3)*(\x+1)*(\x-5)}) |- (5,0) -- (0,0) -- cycle;
\draw[fill=Green,domain=10:14] plot(\x,{((\x-12)^3)/4+2}) |- (14,0) -- (10,0) -- cycle;
\draw[fill=Blue,domain=0:4] plot(\x,{((\x-2)^3)/4+6}) |- (4,8) -- (0,8) -- cycle;
\draw[fill=black,domain=9:14] plot(\x,{(\x-9)*(\x-15)/3+8}) |- (14,8) -- (9,8) -- cycle;
\draw[fill=Green,domain=9:14] (7,4) circle (3);
\end{tikzpicture}%
}

% Layer top over the grid lines
\begin{tikzpicture}[line cap=round,line join=round,x=0.5cm,y=0.5cm,thin,samples=100]
\draw[help lines,step=0.5cm] (0,0) grid (14,8);
\node[inner sep=0pt] (s) at (7,4){\usebox\myimagebox};
% Repeat these to highlight the black outlines again.
\draw[even odd rule,domain=9:14] (0,0) rectangle (14,8) (7,4) circle (3);
\draw[domain=0:5] plot(\x,{-(1/3)*(\x+1)*(\x-5)}) |- (5,0) -- (0,0) -- cycle;
\draw[domain=10:14] plot(\x,{((\x-12)^3)/4+2}) |- (14,0) -- (10,0) -- cycle;
\draw[domain=0:4] plot(\x,{((\x-2)^3)/4+6}) |- (4,8) -- (0,8) -- cycle;
\draw[domain=9:14] plot(\x,{(\x-9)*(\x-15)/3+8}) |- (14,8) -- (9,8) -- cycle;
\draw[domain=9:14] (7,4) circle (3);

\end{tikzpicture}

\end{document}

Answer 2

Here is another method. We fill outside the curves and then inverse clip the central circle.

\documentclass[tikz,border=12pt,svgnames]{standalone}
\tikzset{
  invclip/.style={
    clip,
    insert path={{[reset cm] (-16000pt,-16000pt) rectangle (16000pt,16000pt)}}},
  saveuse path/.code 2 args={
    \pgfkeysalso{#1/.estyle={insert path={#2}}}%
    \global\expandafter\let\csname pgfk@\pgfkeyscurrentpath/.@cmd\expandafter\endcsname 
                           \csname pgfk@\pgfkeyscurrentpath/.@cmd\endcsname
    \pgfkeysalso{#1}}}
\def\invclip#1;{\pgfinterruptboundingbox\path[invclip]#1;\endpgfinterruptboundingbox}

\begin{document}
\begin{tikzpicture}[line cap=round,line join=round,x=0.5cm,y=0.5cm,thin,samples=100,fill opacity=0.5]
\draw[help lines,step=0.5cm] (0,0) grid (14,8);
\draw[fill=red,domain=0:5] plot(\x,{-(1/3)*(\x+1)*(\x-5)})  |- (5,0) -- (0,0) -- cycle;
\draw[fill=Green,domain=10:14]  plot(\x,{((\x-12)^3)/4+2})  |- (14,0) -- (10,0) -- cycle;
\draw[fill=Blue,domain=0:4] plot(\x,{((\x-2)^3)/4+6}) |- (4,8) -- (0,8) -- cycle;
\draw[fill=black,domain=9:14] plot(\x,{(\x-9)*(\x-15)/3+8}) |- (14,8) -- (9,8) -- cycle;

%% inverse clip the circle
\draw [fill=Green,saveuse path={circle}{(7,4) circle (3)}] ;
\invclip [circle];

%% fill the outer parts. Make all colors same. I made them different just for demo
\path[fill=blue!20,domain=0:4]    plot(\x,{((\x-2)^3)/4+6}) -| (9,4);
\path[fill=green!20,domain=10:14]  plot(\x,{((\x-12)^3)/4+2})  -| (5,0);
\path[fill=pink,domain=9:14]   plot(\x,{(\x-9)*(\x-15)/3+8}) |- (9,4);
\path[fill=orange,domain=0:5]   plot(\x,{-(1/3)*(\x+1)*(\x-5)}) |- (0,4);

\end{tikzpicture}
\end{document}

With same colors: