f(x)= a*x + b     if (x < 4.5)
 f(x)= a*x         if (x >= 4.5)

Below is xml for creating a dynamic plot based on this. The work is done in the <script ></script > block that generates the data sets used in the plot.

 <script type="loncapa/perl" >
 $a = &random(2,5,1);
 $b = &random(3,6,1);
 
 for ($x = 0; $x<4.5; $x+= 0.05) {
     push @X, $x;
     push @Y, $a*$x + $b;
 }
 
 for ($x = 4.5; $x<=10; $x+= 0.05) {
     push @X, $x;
     push @Y, $a*$x;
 }
 </script >
 <gnuplot font="medium" width="500" grid="on" height="400" border="on" 
          fgcolor="x000000" alttag="dynamically generated plot" align="center" 
          bgcolor="xffffff" transparent="off" >
     <curve linestyle="lines" pointsize="1" pointtype="1" color="x000000" 
      name="">
         <data >@X</data >
         <data >@Y</data >
     </curve >
 </gnuplot >

A better solution is to use two seperate <curve > statements to plot two separate curves. So on the second piece of the function we use @X2 and @Y2 instead of @X and @Y. Then in the <gnuplot > tag we include a second <curve > sub-tag.

 <script type="loncapa/perl" >
 $a = &random(2,5,1);
 $b = &random(3,6,1);
 
 for ($x = 0; $x<4.5; $x+= 0.05) {
     push @X, $x;
     push @Y, $a*$x + $b;
 }
 
 for ($x = 4.5; $x<=10; $x+= 0.05) {
     push @X2, $x;
     push @Y2, $a*$x;
 }
 </script >
 <gnuplot font="medium" width="500" grid="on" height="400" border="on" 
          fgcolor="x000000" alttag="dynamically generated plot" align="center" 
          bgcolor="xffffff" transparent="off" >
     <curve linestyle="lines" pointsize="1" pointtype="1" color="x000000" 
      name="" >
         <data >@X</data >
         <data >@Y</data >
     </curve >
     <curve linestyle="lines" pointsize="1" pointtype="1" color="x000000" 
      name="" >
         <data >@X2</data >
         <data >@Y2</data >
     </curve >
 </gnuplot >

This is still not a desirable result. Typically one wants an open circle or closed circle defining the domain of each piece. Unfortunately there is no easy way to do this in gnuplot currently. If this effect is desired, perhaps using a static image and a randomlabel problem would be the best approach. A second solution would be to add data which draws a circle to your curves.