\documentclass[11pt]{report}
\input{header}
\pagestyle{empty}
\begin{document}
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    \arraxes(0,0)(-5.5,-2)(5.5,9)
    \psplot[linecolor=blue,linewidth=0.5pt]{-2.1}{2.1}
           {x x mul dup dup mul exch 6 mul sub 8 add}% y = x^4 - 6x^2 + 8
    \psplot[linecolor=olivegreen,linewidth=0.5pt]{-3}{3}
           {x x x mul 1 add div}% y = x / (1 + x^2)
    \psplot[linecolor=orange,linewidth=0.5pt]{-2}{3}{8 x x mul sub}% y = 8 - x^2
    \psplot[linecolor=purple,linewidth=0.5pt]{-3}{0}{0 2 x mul sub}
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    %
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    \arrEmptyVertex(-1.52,-.459){x2}
    \arrEmptyVertex(-1.1,2.2){x0}
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    \arrEmptyVertex(2.77,0.31){x6}
    %
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    \rput(1.4,4){\textcolor{blue}{$c_1$}}
    \rput(3.8,7.7){\textcolor{olivegreen}{$c_2: y = \frac{x}{1 + x^2}$}}
    \rput(-0.9,-0.9){\textcolor{olivegreen}{$c_2$}}
    \rput(3.6,6.9){\textcolor{orange}{$c_3: y = 8 - x^2$}}
    \rput(2.8,3){\textcolor{orange}{$c_3$}}
    \rput(3.7,6.1){\textcolor{purple}{$c_4: y = -2x$}}
    \rput(-2.2,5.3){\textcolor{purple}{$c_4$}}
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}
\end{document}
