\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}