\documentclass{article}
\usepackage[pdftex,active,tightpage]{preview}
\setlength\PreviewBorder{2mm}

\usepackage{amssymb,amsmath}
\usepackage{pgfplots}
\usepackage{tikz}
\usetikzlibrary{arrows, positioning, calc, intersections, decorations.pathreplacing}

\newcommand\tikzmark[1]{%
  \tikz[overlay,remember picture,baseline] \coordinate [anchor=base] (#1);}

\newcommand\DrawBrace[3]{%
  \draw [ultra thick, decorate,decoration={brace,amplitude=5pt,mirror,raise=2pt}]
    (#1) -- (#2) node [midway,xshift=23pt] {\Huge $\displaystyle #3$};
}

\begin{document}

\begin{preview}
\begin{tikzpicture}
    \pgfplotsset{
        right segments/.code={\pgfmathsetmacro\rightsegments{#1}},
        right segments=3,
        right/.style args={#1:#2}{
            ybar interval,
            domain=#1+((#2-#1)/\rightsegments):#2+((#2-#1)/\rightsegments),
            samples=\rightsegments+1,
            x filter/.code=\pgfmathparse{\pgfmathresult-((#2-#1)/\rightsegments)}
        }
    }

    \pgfplotsset{
        left segments/.code={\pgfmathsetmacro\leftsegments{#1}},
        left segments=3,
        left/.style args={#1:#2}{
            ybar interval,
            domain=#1:#2,
            samples=\leftsegments+1,
            x filter/.code=\pgfmathparse{\pgfmathresult}
        }
    }

    \begin{axis}[
        axis lines=middle,
        width=15cm, height=15cm,     % size of the image
        grid = major,
        grid style={dashed, gray!30},
        xmin= 0, % start the diagram at this x-coordinate
        xmax= 3, % end   the diagram at this x-coordinate
        ymin= 0, % start the diagram at this y-coordinate
        ymax= 9, % end   the diagram at this y-coordinate
        axis background/.style={fill=white},
        ylabel=y,
        xlabel=x,
        tick align=outside,
        tension=0.08,
        legend style={at={(0.25,0.91)}, anchor=north}]
      \addplot[domain=0:3,ultra thin, red, samples=500] {x};
      \addplot[domain=0:3,very thin, red, samples=500] {x^2};
      \addplot[domain=0:3,thin, red, samples=500] {x^3};
      \addplot[domain=0:3,semithick, red, samples=500] {x^4};
      \addplot[domain=0:3,thick, red, samples=500] {x^5};
      \addplot[domain=0:3,very thick, red, samples=500] {e^x};
      \addplot[domain=0:3,ultra thick, red, samples=500] {sqrt(x)};

      \legend{ultra thin, very thin, thin, semithick, thick, very thick, ultra thick};
    \end{axis}
\end{tikzpicture}
\end{preview}
\end{document}