document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1092395>Question 1092395</A>: <I><SPAN STYLE=\"word-break: break-all;\"> graph of function<BR>\n" );
document.write( "y= (|x-3|+|x+1|)/(|x+3|+|x-1|) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #707005 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=707005\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br>The behavior of the function changes each time we pass a point where one of the expressions in absolute value symbols is equal to 0.<br><BR>\n" );
document.write( "For checking the graph we come up with, it is useful to evaluate the function at each of those points.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-3%29+=+%286%2B2%29%2F%280%2B4%29+=+8%2F4+=+2 ALIGN=MIDDLE  ALT=\"f%28-3%29+=+%286%2B2%29%2F%280%2B4%29+=+8%2F4+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-1%29+=+%284%2B0%29%2F%282%2B2%29+=+4%2F4+=+1 ALIGN=MIDDLE  ALT=\"f%28-1%29+=+%284%2B0%29%2F%282%2B2%29+=+4%2F4+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29+=+%282%2B2%29%2F%284%2B0%29+=+4%2F4+=+1 ALIGN=MIDDLE  ALT=\"f%281%29+=+%282%2B2%29%2F%284%2B0%29+=+4%2F4+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%283%29+=+%280%2B4%29%2F%286%2B2%29+=+4%2F8+=+.5 ALIGN=MIDDLE  ALT=\"f%283%29+=+%280%2B4%29%2F%286%2B2%29+=+4%2F8+=+.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So the places where the behavior of the graph changes are (-3,2), (-1,1), (1,1), and (3,.5)<br><BR>\n" );
document.write( "(1) For x < -3...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-x%2B3%29%2B%28-x-1%29%29%2F%28%28-x-3%29%2B%28-x%2B1%29%29+=+%28-2x%2B2%29%2F%28-2x-2%29+=+%282x-2%29%2F%282x%2B2%29+=+1-4%2F%282x%2B2%29+=+1-2%2F%28x%2B1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This is a segment of a branch of a hyperbola with horizontal asymptote y=1 and vertical asymptote at x = -1.<br><BR>\n" );
document.write( "(2) For -3 < x < -1...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-x%2B3%29%2B%28-x-1%29%29%2F%28%28x%2B3%29%2B%28-x%2B1%29%29+=+%28-2x%2B2%29%2F4+=+%28-1%2F2%29x+%2B+1%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This is of course a segment of a line.<br><BR>\n" );
document.write( "(3) For -1 < x < 1...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-x%2B3%29%2B%28x%2B1%29%29%2F%28%28x%2B3%29%2B%28-x%2B1%29%29+=+4%2F4+=+1%29 ALIGN=MIDDLE  ALT=\"%28%28-x%2B3%29%2B%28x%2B1%29%29%2F%28%28x%2B3%29%2B%28-x%2B1%29%29+=+4%2F4+=+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And this is clearly a constant function.<br><BR>\n" );
document.write( "(4) For 1 < x < 3...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-x%2B3%29%2B%28x%2B1%29%29%2F%28%28x%2B3%29%2B%28x-1%29%29+=+4%2F%282x%2B2%29+=+2%2F%28x%2B1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This is a segment of a branch of a hyperbola with horizontal asymptote y=0 and vertical asymptote at x = -1.<br><BR>\n" );
document.write( "(5) And for x > 3...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x-3%29%2B%28x%2B1%29%29%2F%28%28x%2B3%29%2B%28x-1%29%29+=+4%2F%282x%2B2%29+=+2%2F%28x%2B1%29%282x-2%29%2F%282x%2B2%29+=+1-4%2F%282x%2B2%29+=+1-2%2F%28x%2B1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And this is part of the other branch of the same hyperbola as in (1).<br><BR>\n" );
document.write( "Here is a graph of the function.  Sorry; I haven't yet figured out how to draw the graph of a piecewise function.<br><BR>\n" );
document.write( "(1) red line from (-infinity,1) to (-3,2)<BR>\n" );
document.write( "(2) green line from (-3,2) to (-1,1)<BR>\n" );
document.write( "(3) blue line from (-1,1) to (1,1)<BR>\n" );
document.write( "(4) purple line from (1,1) to (3,.5)<BR>\n" );
document.write( "(5) red line again from (3,.5) to (infinity, 1)<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C200%2C-10%2C10%2C0%2C3%2C1-2%2F%28x%2B1%29%2C-0.5x%2B.5%2C1%2C2%2F%28x%2B1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( " </SPAN>\n" );
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