Elegant hash areas for mathematical graphics...

Question

Sometimes on a graph you need to hash out areas in order to show vividly what the domains are. Is there an elegant way to create these hash areas using an instruction similar to the rectangle one? Would it be possible to hash out the interior of any shape? And it would be awesome if the hash would allow automatically for gaps were notation needed be over-imposed (like for x and 1 in my MWE)... I am aware of the pattern=north east lines, but that one is too rigid, I cannot control the distance between the lines. My MWE is kinda verbose, sorry...

\documentclass{standalone}
\usepackage{pgfplots}
\usetikzlibrary{arrows}

\begin{document}

\resizebox{7.0cm}{!}{
\begin{tikzpicture}[>=triangle 45]% functia PUTERE 2^x
%\clip (-0.5,-0.5) rectangle (4.0,4.0);
% grid
%\draw[step = 0.1cm, gray, ultra thin] (-5.0, -1.0) grid (4.0, 0.0);
%
% HASH lines for y < 0
%
\draw[red, line width=1.00pt] (-5.0, -0.1) -- (-4.9, 0.0);
\draw[red, line width=1.00pt] (-5.0, -0.5) -- (-4.5, 0.0);
\draw[red, line width=1.00pt] (-5.0, -1.0) -- (-4.0, 0.0);
\draw[red, line width=1.00pt] (-4.5, -1.0) -- (-3.5, 0.0);
\draw[red, line width=1.00pt] (-4.0, -1.0) -- (-3.0, 0.0);
\draw[red, line width=1.00pt] (-3.5, -1.0) -- (-2.5, 0.0);
\draw[red, line width=1.00pt] (-3.0, -1.0) -- (-2.0, 0.0);
\draw[red, line width=1.00pt] (-2.5, -1.0) -- (-1.5, 0.0);
\draw[red, line width=1.00pt] (-2.0, -1.0) -- (-1.0, 0.0);
\draw[red, line width=1.00pt] (-1.5, -1.0) -- (-0.5, 0.0);
\draw[red, line width=1.00pt] (-1.0, -1.0) -- (0.0, 0.0);
\draw[red, line width=1.00pt] (-0.5, -1.0) -- (0.5, 0.0);
% gap for x notation
\draw[red, line width=1.00pt] (0.0, -1.0) -- (0.4, -0.6);
\draw[red, line width=1.00pt] (0.7, -0.3) -- (1.0, 0.0);
%
\draw[red, line width=1.00pt] (0.5, -1.0) -- (1.5, 0.0);
\draw[red, line width=1.00pt] (1.0, -1.0) -- (2.0, 0.0);
\draw[red, line width=1.00pt] (1.5, -1.0) -- (2.5, 0.0);
\draw[red, line width=1.00pt] (2.0, -1.0) -- (3.0, 0.0);
\draw[red, line width=1.00pt] (2.5, -1.0) -- (3.5, 0.0);
\draw[red, line width=1.00pt] (3.0, -1.0) -- (4.0, 0.0);
\draw[red, line width=1.00pt] (3.5, -1.0) -- (4.0, -0.5);
\draw[red, line width=1.00pt] (3.9, -1.0) -- (4.0, -0.9);
%
% 
\coordinate (OR) at (0.00, 0.00);
\coordinate (LX) at (-5.00, 0.00); % left x
\coordinate (RX) at (4.00, 0.00); % right x
\coordinate (BY) at (0.00, -1.00); % bottom y
\coordinate (TY) at (0.00, 8.00);
%
% axa 0x
%
\draw[->][line width=1.00pt] (LX) -- (RX);
\node[violet] at (3.8,-0.4) {\textbf{\textit{x}}};
%
% axa 0y
%
\draw[->][line width=1.00pt] (BY) -- (TY);
\node[right,violet] at (0.2, 8.0) {{\boldmath$a^{\raisebox{+4.0pt}{$\scriptstyle{x}$}}$}};
\node[right,violet] at (0.1, 7.3) {{\boldmath$(a \,=\, 2)$}};
%
% ORIGINE
% points on the graphic
\filldraw [red] (OR) circle(2pt);
\node[violet] at (-0.2,-0.3) {\textbf{\textit{0}}};
%
% COORDONATE -- numerotare axa Ox POZITIVA
%
\draw[violet,line width=1pt] (0.50, -0.10) -- (0.50, 0.10); % x = 0.5
\node[violet] at (0.50, -0.40) {\textbf{\textit{x}}}; 
\draw[violet,line width=0.75pt,dotted] (0.50, 0.00) -- (0.50, 1.4142);
\filldraw [violet] (0.50, 1.4142) circle(2pt);
\node[above,violet] at (0.50, 1.4142) {\textbf{\textit{a\textsuperscript{x}}}}; 
%
\draw[line width=1pt] (1.00, -0.10) -- (1.00, 0.10); % x1
\node at (1.00, -0.40) {$\scriptstyle{1}$};
\draw[line width=1pt] (2.00, -0.10) -- (2.00, 0.10); % x2
\node at (2.00, -0.40) {$\scriptstyle{2}$};
\draw[line width=1pt] (3.00, -0.10) -- (3.00, 0.10); % x3
\node at (3.00, -0.40) {$\scriptstyle{3}$};
%
% COORDONATE -- numerotare axa Ox NEGATIVA
%
\draw[line width=1pt] (-1.00, -0.10) -- (-1.00, 0.10); % -x1
\node at (-1.10, -0.40) {$\scriptstyle{-1}$};
\draw[line width=1pt] (-2.00, -0.10) -- (-2.00, 0.10); % -x2
\node at (-2.10, -0.40) {$\scriptstyle{-2}$};
\draw[line width=1pt] (-3.00, -0.10) -- (-3.00, 0.10); % -x3
\node at (-3.10, -0.40) {$\scriptstyle{-3}$};
\draw[line width=1pt] (-4.00, -0.10) -- (-4.00, 0.10); % -x4
\node at (-4.10, -0.40) {$\scriptstyle{-4}$};
%
%
% COORDONATE -- numerotare axa Oy POZITIVA
%
\draw[line width=1.00pt] (-0.1, 1.00) -- (0.10, 1.00); % y1 
\node at (-0.40, 1.00) {$\scriptstyle{1}$};
\draw[line width=1.00pt] (-0.1, 2.00) -- (0.10, 2.00); % y2 
\node at (-0.40, 2.00) {$\scriptstyle{2}$};
\draw[line width=1.00pt] (-0.1, 3.00) -- (0.10, 3.00); % y3
\node at (-0.40, 3.00) {$\scriptstyle{3}$};
\draw[line width=1.00pt] (-0.1, 4.00) -- (0.10, 4.00); % y4 
\node at (-0.40, 4.00) {$\scriptstyle{4}$};
\draw[line width=1.00pt] (-0.1, 5.00) -- (0.10, 5.00); % y5 
\node at (-0.40, 5.00) {$\scriptstyle{5}$};
\draw[line width=1.00pt] (-0.1, 6.00) -- (0.10, 6.00); % y6 
\node at (-0.40, 6.00) {$\scriptstyle{6}$};
\draw[line width=1.00pt] (-0.1, 7.00) -- (0.10, 7.00); % y7 
\node at (-0.40, 7.00) {$\scriptstyle{7}$};
%
% COORDONATE -- numerotare axa Oy NEGATIVA
%
% GRAFICUL functiei 2^x
%
\draw[violet, line width=1.75pt, domain=-5.00:3.00] plot[smooth](\x, {2^(\x)});
%
% filled circles
%
\filldraw [blue] (0.0,1.0) circle(2pt);
%
\end{tikzpicture}
}
%
\hspace*{1.0cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\resizebox{8.0cm}{!}{
\begin{tikzpicture}[>=triangle 45]%functia LOGARITM ln x
% grid
%\draw[step = 0.1cm, gray, ultra thin] (-1.0, 0.0) grid (0.0, 2.0);
%
% HASH lines for x < 0
%
\draw[red, line width=1.00pt] (-1.0, 2.9) -- (-0.9, 3.0);
\draw[red, line width=1.00pt] (-1.0, 2.5) -- (-0.5, 3.0);
\draw[red, line width=1.00pt] (-1.0, 2.0) -- (0.0, 3.0);
\draw[red, line width=1.00pt] (-1.0, 1.5) -- (0.0, 2.5);
\draw[red, line width=1.00pt] (-1.0, 1.0) -- (0.0, 2.0);
% gap for 1
\draw[red, line width=1.00pt] (-1.0, 0.5) -- (-0.5, 1.0);
\draw[red, line width=1.00pt] (-0.3, 1.2) -- (0.0, 1.5);
%
\draw[red, line width=1.00pt] (-1.0, 0.0) -- (0.0, 1.0);
\draw[red, line width=1.00pt] (-1.0, -0.5) -- (0.0, 0.5);
\draw[red, line width=1.00pt] (-1.0, -1.0) -- (0.0, 0.0);
\draw[red, line width=1.00pt] (-1.0, -1.5) -- (0.0, -0.5);
\draw[red, line width=1.00pt] (-1.0, -2.0) -- (0.0, -1.0);
\draw[red, line width=1.00pt] (-1.0, -2.5) -- (0.0, -1.5);
\draw[red, line width=1.00pt] (-1.0, -3.0) -- (0.0, -2.0);
\draw[red, line width=1.00pt] (-1.0, -3.5) -- (0.0, -2.5);
\draw[red, line width=1.00pt] (-1.0, -4.0) -- (0.0, -3.0);
\draw[red, line width=1.00pt] (-1.0, -4.5) -- (0.0, -3.5);
\draw[red, line width=1.00pt] (-1.0, -5.0) -- (0.0, -4.0);
\draw[red, line width=1.00pt] (-1.0, -4.5) -- (0.0, -3.5);
\draw[red, line width=1.00pt] (-1.0, -5.0) -- (0.0, -4.0);
\draw[red, line width=1.00pt] (-0.5, -5.0) -- (0.0, -4.5);
\draw[red, line width=1.00pt] (-0.1, -5.0) -- (0.0, -4.9);
%
% LOGARITM function
%
%\clip (-0.5,-0.5) rectangle (4.0,4.0);
%
% grid, drawn with a step in a specified rectangle
%\draw[step = 0.1cm, gray, ultra thin] (-1.0, -5.0) grid (0.0,3.0);
%
%\draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
%
%
%\begin{comment}
% 
\coordinate (OR) at (0.00, 0.00);
\coordinate (LX) at (-1.00, 0.00);
\coordinate (RX) at (8.00, 0.00);
\coordinate (BY) at (0.00, -5.00);
\coordinate (TY) at (0.00, 3.00);
%
% axa 0x
%
\draw[->][line width=1.00pt] (LX) -- (RX);
\node[blue] at (7.8,-0.4) {\textbf{\textit{x}}};
%
% axa 0y
%
\draw[->][line width=1.00pt] (BY) -- (TY);
\node[right,blue] at (0.2, 2.8) {\textbf{\textit{ln x}}};
%
% ORIGIN
% points on the graphic
\filldraw [red] (OR) circle(2pt);
\node[red] at (0.2,-0.3) {\textbf{\textit{0}}};
%
% COORDONATE -- numerotare axa Ox POZITIVA
%
\draw[line width=1pt] (1.00, -0.10) -- (1.00, 0.10); % x1
\node at (1.00, -0.40) {$\scriptstyle{1}$};
\draw[line width=1pt] (2.00, -0.10) -- (2.00, 0.10); % x2
\node at (2.00, -0.40) {$\scriptstyle{2}$};
\draw[line width=1pt] (3.00, -0.10) -- (3.00, 0.10); % x3
\node at (3.00, -0.40) {$\scriptstyle{3}$};
\draw[line width=1pt] (4.00, -0.10) -- (4.00, 0.10); % x4
\node at (4.00, -0.40) {$\scriptstyle{4}$};
\draw[line width=1pt] (5.00, -0.10) -- (5.00, 0.10); % x5
\node at (5.00, -0.40) {$\scriptstyle{5}$};
\draw[line width=1pt] (6.00, -0.10) -- (6.00, 0.10); % x6
\node at (6.00, -0.40) {$\scriptstyle{6}$};
\draw[line width=1pt] (7.00, -0.10) -- (7.00, 0.10); % x7
\node at (7.00, -0.40) {$\scriptstyle{7}$};
%
% COORDONATE -- numerotare axa Ox NEGATIVA
%
% COORDONATE -- numerotare axa Oy POZITIVA
%
\draw[magenta, line width=1.00pt] (-0.1, 1.00) -- (0.10, 1.00); % y1 
\node[magenta] at (-0.40, 1.00) {$1$};
\draw[line width=1.00pt] (-0.1, 2.00) -- (0.10, 2.00); % y2 
\node at (-0.40, 2.00) {$\scriptstyle{2}$};
%
% COORDONATE -- numerotare axa Oy NEGATIVA
%
\draw[line width=1.00pt] (-0.1, -1.00) -- (0.10, -1.00); % y1 
\node at (-0.40, -1.00) {$\scriptstyle{-1}$};
\draw[line width=1.00pt] (-0.1, -2.00) -- (0.10, -2.00); % y2 
\node at (-0.40, -2.00) {$\scriptstyle{-2}$};
\draw[line width=1.00pt] (-0.1, -3.00) -- (0.10, -3.00); % y3
\node at (-0.40, -3.00) {$\scriptstyle{-3}$};
\draw[line width=1.00pt] (-0.1, -4.00) -- (0.10, -4.00); % y4 
\node at (-0.40, -4.00) {$\scriptstyle{-4}$};
\draw[line width=1.00pt] (-0.1, -5.00) -- (0.10, -5.00); % y5 
\node at (-0.40, -5.00) {$\scriptstyle{-5}$};
%
% COORDINATES for drawing
%
%
\coordinate (EE) at (2.7182, 0.00);
\coordinate (PI) at (3.1415, 0.00);
%
% GRAFICUL functiei LOGARITM
%
\draw[blue, line width=1.75pt, domain=0.01:7.00, samples=500] plot[smooth](\x, {ln(\x)});
%
% filled circles
%
\filldraw [red] (1.00, 0.00) circle(2pt);
%
% punctul e
%
\filldraw [magenta] (EE) circle(2pt);
\node[magenta] at (2.7182, -0.30) {\textbf{\textit{e}}};
\draw[magenta, dotted, line width=1.00pt] (2.7182, 0.00) -- (2.7182, 1.00); % e
\draw[magenta, dotted, line width=1.00pt] (0.00, 1.00) -- (2.7182, 1.00); % e
\filldraw [magenta] (2.7182, 1.00) circle(2pt);
%
\end{tikzpicture}
}

\end{document}

Answer 1

Next code shows how to use define a pattern with user defined \hatchthickness and \hatchdistance. This pattern was taken from using pattern inside tikz shapes with dropped shadows

This pattern is used to fill a rectangular node which is drawn in background layer:

\begin{scope}[on background layer]
\node[fit={(-5,0) (4,-1)}, inner sep=0pt, Pattern] {};
\end{scope}

The pattern doesn't consider holes but you can fill=white all nodes which need to be readable over it:

\node[violet,fill=white] at (3.8,-0.4) {\textbf{\textit{x}}};

Your code has been reduced with the pattern definition, but also using some \foreach loops to draw ticks. And I think it could be reduced even more if you used pgfplots which is loaded but never used.

Next code shows only left figure. Right one can be easily obtained applying similar tools.

\documentclass[border=3mm,tikz]{standalone}
\usepackage{pgfplots}
\usetikzlibrary{arrows,patterns,backgrounds,fit}

\makeatletter
\tikzset{% customization of pattern
         % based on <m.wibrow@gm...> - 2013-03-24 07:20: 
        hatch distance/.store in=\hatchdistance,
        hatch distance=5pt,
        hatch thickness/.store in=\hatchthickness,
        hatch thickness=5pt
        }
\pgfdeclarepatternformonly[\hatchdistance,\hatchthickness]{north east hatch}% name
    {\pgfqpoint{-1pt}{-1pt}}% below left
    {\pgfqpoint{\hatchdistance}{\hatchdistance}}% above right
    {\pgfpoint{\hatchdistance-1pt}{\hatchdistance-1pt}}%
    {
        \pgfsetcolor{\tikz@pattern@color}
        \pgfsetlinewidth{\hatchthickness}
        \pgfpathmoveto{\pgfqpoint{0pt}{0pt}}
        \pgfpathlineto{\pgfqpoint{\hatchdistance}{\hatchdistance}}
        \pgfusepath{stroke}
    }
\makeatother

\tikzset{Pattern/.style={pattern=north east hatch, 
         pattern color=red, hatch distance=3mm, 
         hatch thickness=1pt}}

\begin{document}

%\resizebox{7.0cm}{!}{
\begin{tikzpicture}[>=triangle 45]% functia PUTERE 2^x
%\clip (-0.5,-0.5) rectangle (4.0,4.0);
% grid
%\draw[step = 0.1cm, gray, ultra thin] (-5.0, -1.0) grid (4.0, 0.0);

%
% 
\coordinate (OR) at (0.00, 0.00);
\coordinate (LX) at (-5.00, 0.00); % left x
\coordinate (RX) at (4.00, 0.00); % right x
\coordinate (BY) at (0.00, -1.00); % bottom y
\coordinate (TY) at (0.00, 8.00);
%
% axa 0x
%
\draw[->][line width=1.00pt] (LX) -- (RX);
\node[violet,fill=white] at (3.8,-0.4) {\textbf{\textit{x}}};
%
% axa 0y
%
\draw[->][line width=1.00pt] (BY) -- (TY);
\node[right,violet] at (0.2, 8.0) {{\boldmath$a^{\raisebox{+4.0pt}{$\scriptstyle{x}$}}$}};
\node[right,violet] at (0.1, 7.3) {{\boldmath$(a \,=\, 2)$}};
%
% ORIGINE
% points on the graphic
\filldraw [red] (OR) circle(2pt);
\node[violet,fill=white] at (-0.2,-0.3) {\textbf{\textit{0}}};
%
% COORDONATE -- numerotare axa Ox POZITIVA
%
\draw[violet,line width=1pt] (0.50, -0.10) -- (0.50, 0.10); % x = 0.5
\node[violet,fill=white] at (0.50, -0.40) {\textbf{\textit{x}}}; 
\draw[violet,line width=0.75pt,dotted] (0.50, 0.00) -- (0.50, 1.4142);
\filldraw [violet] (0.50, 1.4142) circle(2pt);
\node[above,violet] at (0.50, 1.4142) {\textbf{\textit{a\textsuperscript{x}}}}; 
%

\foreach \i in {-4,...,-1,1,2,3}{
 \draw[line width=1pt] (1.00*\i, -0.10) -- ++(90:0.20); % x1
 \node[fill=white] at (1.00*\i, -0.40) {$\scriptstyle{\i}$};}

%
% COORDONATE -- numerotare axa Oy POZITIVA
%
\foreach \i in {1,...,7}{
    \draw[line width=1.00pt] (-0.1, \i) -- ++(0:0.20); % y1 
    \node at (-0.40, \i) {$\scriptstyle{\i}$};
    }
%
% COORDONATE -- numerotare axa Oy NEGATIVA
%
% GRAFICUL functiei 2^x
%
\draw[violet, line width=1.75pt, domain=-5.00:3.00] plot[smooth](\x, {2^(\x)});
%
% filled circles
%
\filldraw [blue] (0.0,1.0) circle(2pt);
%
\begin{scope}[on background layer]
\node[fit={(-5,0) (4,-1)}, inner sep=0pt, Pattern] {};
\end{scope}

\end{tikzpicture}

Update

Thank you to Gonzalo Medina, we have a pgfplots version for this figure:

\documentclass[border=3mm,tikz]{standalone}
\usepackage{pgfplots}
\usetikzlibrary{arrows,patterns,backgrounds,fit}

\makeatletter
\tikzset{% customization of pattern
         % based on <m.wibrow@gm...> - 2013-03-24 07:20:
        hatch distance/.store in=\hatchdistance,
        hatch distance=5pt,
        hatch thickness/.store in=\hatchthickness,
        hatch thickness=5pt
        }
\pgfdeclarepatternformonly[\hatchdistance,\hatchthickness]{north east hatch}% name
    {\pgfqpoint{-1pt}{-1pt}}% below left
    {\pgfqpoint{\hatchdistance}{\hatchdistance}}% above right
    {\pgfpoint{\hatchdistance-1pt}{\hatchdistance-1pt}}%
    {
        \pgfsetcolor{\tikz@pattern@color}
        \pgfsetlinewidth{\hatchthickness}
        \pgfpathmoveto{\pgfqpoint{0pt}{0pt}}
        \pgfpathlineto{\pgfqpoint{\hatchdistance}{\hatchdistance}}
        \pgfusepath{stroke}
    }
\makeatother

\tikzset{Pattern/.style={pattern=north east hatch,
         pattern color=#1, hatch distance=3mm,
         hatch thickness=1pt}}

\begin{document}

\begin{tikzpicture}[>=triangle 45]
\begin{axis}[
  axis lines=middle,
  ymax=8,
  ymin=-1,
  xmax=3.5,
  clip=false,
  xtick={-4,-3,...,3},
  ytick={1,2,...,8},
  x tick label style={fill=white}
  ]
\addplot[violet, line width=1.75pt, domain=-5.00:3.00] {2^(x)};

\node[violet,fill=white]
  at (axis cs:3.8,-0.4) {\textbf{\textit{x}}};
\node[right,violet]
  at (axis cs:0.2, 8.0)
  {{\boldmath$a^{x}$}};
\node[right,violet]
  at (axis cs:0.1, 7.3) {{\boldmath$(a \,=\, 2)$}};
\filldraw[red] (axis cs:0,0) circle(2pt);
\draw[violet,line width=0.75pt,dotted]
  (axis cs:0.50, 0.00) -- (axis cs:0.50, 1.4142);
\filldraw[violet] (axis cs:0.50, 1.4142) circle(2pt)
  node[above,violet]
    at (axis cs:0.50, 1.4142)
    {\textbf{\textit{a\textsuperscript{x}}}};
\filldraw[blue] (axis cs:0.0,1.0) circle(2pt);

\begin{scope}[on background layer]
\node[fit={(axis cs:-5,0) (axis cs:3.5,-1.3)}, inner sep=0pt, Pattern=cyan] {};
\end{scope}
\end{axis}
\end{tikzpicture}

\end{document}

which produces:

Answer 2

I took the liberty to draw one of the examples with the mfpic package, which comes with a macro precisely built for those kind of problems:

\thatch[<hatchsp>,<angle>][<color>]<closed path> 

It fills any closed path with lines at <angle>degrees, separated from each other by a distance of hatchsp. Another macro, \hatchwd{} macro specifies the width of those lines. There is also two other hatching macros as variants of \thatch with predefined angles, \rhatch (going from lower left to upper right) and \lhatch (going from lower right to upper left).

In this example all the hatching work is done by the following instructions:

\hatchwd{1pt}
\rhatch[3mm][Cyan]\rect{(\xmin, \ymin), (\xmax, 0)}

The complete code:

\documentclass{scrartcl}
\usepackage{multido, xcolor}
\usepackage[metapost, clip, overlaylabels]{mfpic}
    \setlength{\mfpicunit}{1cm}
    \opengraphsfile{\jobname}
\begin{document}
\begin{mfpic}[1]{-7}{3.9}{-1.5}{8.9}
    % Function
    \penwd{1pt}
    \fdef{f}{x}{2**x}
   \draw[magenta]\function{\xmin, \xmax, 0.05}{f(x)}    
   % Hatching
   \hatchwd{1pt}
   \rhatch[3mm][Cyan]\rect{(\xmin, \ymin), (\xmax, 0)}
   % Axes and marks
   \penwd{.5pt}
   \doaxes{xy}
   \xmarks{ceiling(\xmin) upto -1}\xmarks{1 upto floor(\xmax)}
   \ymarks{ceiling(\ymin) upto -1}\ymarks{2 upto floor(\ymax)}
   \pointcolor{red}\point[4pt]{(0, 0)}
   \pointcolor{blue}\point[4pt]{(0, 1)}
   % Labels
   \setrender{\gclear}
   \tlpointsep{3pt}
    \tlabelrect[][tr](0, 0){$O$} 
    \tlabelrect[][tc](\xmax, 0){$x$} 
    \tlabel[cr](0, \ymax){$y$}
    \tlpointsep{6pt}
    \tlpathsep{1pt}
    \multido{\i=\xmin+1}{-\xmin}{\tlabelrect[][tc](\i, 0){$\i$}}
    \multido{\i=1+1}{\xmax}{\tlabelrect[][tc](\i, 0){$\i$}}
    \multido{\i=1+1}{\ymax}{\tlabel[cr](0, \i){$\i$}}
    \tlabelrect[][cr](0, -1){$-1$}
    \drawcolor{magenta}
    \dashed\lines{(0.5, 0), (0.5, 1.414)}
    \tlabel[cr](3, 8){\textcolor{magenta}{$a^x (a=2)$}}
    \tlabel[bc](0.5, 1.414){\textcolor{magenta}{$a^x$}}
    \tlabelrect[][tc](0.5, 0){\textcolor{magenta}{$x$}}
\end{mfpic}
\closegraphsfile
\end{document}

To be compiled with LaTeX first (no matter the engine), then the resulting .mp file with MetaPost, and then the .tex file again with LaTeX.