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\draw[thick,domain=-2:0] plot(\x,{0.5*\x^3-1.5*\x^2+2});

\draw[smooth,samples=100,domain=-4.0:0.0] plot(\x,{-3.5*ln(-\x+1)});

\draw[thick,samples=100,domain=-2:0] plot{\x,-0.5*\x^3-1.5*\x^2+2};

\documentclass{minimal}
\usepackage{tikz}

\begin{document}
  \begin{tikzpicture}[domain=-5:3]
    \draw[very thin,color=gray] (0.0,0.0) grid (0,0);
    \draw[->] (-4.5,0) -- (4.0,0) node[right] {$x$};
    \draw[->] (0,-4.0) -- (0,3.0) node[above] {$\mu(x)$};
    \draw[green,domain=0.0:3.9] plot (\x,{1.3*ln(\x+1)});
    \draw (2,1) node[right,green] {$m(c)>m(r)$};
    \draw[red,smooth,samples=100,domain=-2.0:0.0] plot(\x,{-3.5*ln(-\x+1)});
    \draw (-1,-2) node[red,left] {$m(c)<m(r)$};
  \end{tikzpicture} 
\end{document}

\begin{figure}[H]
   \centering
\begin{tikzpicture}[domain=-5:3]
\draw[very thin,color=gray] (0.0,0.0) grid (0,0);

\draw[->] (-4.5,0) -- (4.0,0) node[right] {$x$};
\draw[->] (0,-4.0) -- (0,3.0) node[above] {$\mu(x)$};

\draw[color=green,domain=0.0:3.9] plot (\x,{1.3*ln(\x+1)}) node[below, midway] {$m(c)>m(r)$};
\draw[color=red,smooth,samples=100,domain=-2.0:0.0] plot(\x,{-3.5*ln(-\x+1)}) node[above, midway] {$m(c)<m(r)$};;
\end{tikzpicture}

   \caption[Normale Figure Umgebung]{Dies ist eine lange Abbildungungsbeschriftung. Dies ist eine lange Abbildungungsbeschriftung. Dies ist eine lange Abbildungungsbeschriftung.}
\end{figure}

\draw[color=red,smooth,samples=100,domain=-3.0:0.0] plot(\x,{-3.5*ln(-\x+1)})  node[left] {$f(x)=x$};;

\draw[smooth,samples=100,domain=-4.0:0.0] plot(\x,{-3.5*ln(-\x+1)});

\documentclass{minimal}
\usepackage{pgfplots}

\begin{document}
  \begin{tikzpicture}
    \begin{axis}
      \addplot[domain=0:4] {1.3*ln(x+1)};
      \addplot[domain=-4:0] {-3.5*ln(-x+1)};
    \end{axis}
  \end{tikzpicture}
\end{document}

\draw[smooth,samples=100,domain=0.0:4.0] plot(\x,{1.3*ln(\x+1)});
\draw[smooth,samples=100,domain=-4.0:0.0] plot(\x,{0-(3.5)*ln(-(\x)+1)});