You mention that you've been tasked with drawing simple 3D structures. For that, a visually intuitive tool is Google Sketchup. 3D geometry for cubes is obtained by extruding (using the Push/Pull tool) a square, and model components can be intersected with one another.
![enter image description here](https://i.stack.imgur.com/UEYR4.gif)
Geometric perspective to the desired viewpoint is easily achieved via the orbiting tool. Print to PDF provides a means to include the resulting 3D structure to be included into your LaTeX document:
![enter image description here](https://i.stack.imgur.com/SeD6g.png)
More complex figures and intersections are also possible using basic constructions of rectangles/circles and extrusions in 3D. Here the two pieces from the intersected cube has been separated:
![enter image description here](https://i.stack.imgur.com/veYEr.png)
Your path line
is to short, it doesn't cross path curve
but only touch it. It seems that this make at older version of pgfplots
your problem, however with recent version (1.14) I can't reproduce your problem. Anyway, to be on safe side, I suggest some changes in your MWE. They are indicated by % <--
in code:
\documentclass[tikz,
border=3mm,
]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.14, set layers}
\usetikzlibrary{intersections}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
domain=0.1:6,
axis lines = left,
xlabel = $x$,
ylabel = {$y$},
xmin=0.1, xmax=6,
ymin=0.1, ymax=1,
scaled ticks=false,
xticklabel=\empty, yticklabel=\empty,
xtick=\empty, ytick=\empty,
every axis plot post/.append style={very thick, smooth,}
]
\addplot[name path=A,black] {(3*x-2+0.4*4.4)/(3*x+0.4*4.4)}
node[pos=0.8,sloped,yshift=6pt]{$y_{0}$};
\addplot[blue] {(3*x-2)/(3*x)}
node[pos=0.95,sloped,yshift=-6pt]{$y_{1}$};
\path[name path=B] (2,0) -- (2,2); % <-- added, this path cross pat A
\draw[draw,name intersections={of=A and B, by={a}},fill=white]% <-- intersection i named by "a"
(a) circle (2pt);
\pgfonlayer{pre main}
\coordinate (b) at (0.1,0.1);
\draw[red, densely dashed] (a) -- (a |- b) node[below,text=black] {$x_i$};% <-- instead of addplot for path "line"
\endpgfonlayer
\end{axis}
\end{tikzpicture}
\end{document}
![enter image description here](https://i.stack.imgur.com/amzT1.png)
Best Answer
pst-solides3d
seems a good option here. In the PS-Tricksgallery of 3D examples
you have a nice starting point by Manuel Luque:The resulting image (taken directly from the PSTricks homepage):