document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=439239>Question 439239</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help...<BR>\n" );
document.write( "Sketch the graph of Arccos x if Cos x has a domain of 0 &#8804; x &#8804; &#960;  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #303531 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=303531\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "here is y = cos^-1x = arccosx and its graph:\r<BR>\n" );
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document.write( "open this image:\r<BR>\n" );
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document.write( "http://img23.imageshack.us/f/arccos3.gif/\r<BR>\n" );
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document.write( "Since {y = cos^-1(x) is the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=inverse ALIGN=MIDDLE  ALT=\"inverse\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+cos+x ALIGN=MIDDLE  ALT=\"y+=+cos+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the function y = cos^-1(x)<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=if ALIGN=MIDDLE  ALT=\"if\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=only ALIGN=MIDDLE  ALT=\"only\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+if ALIGN=MIDDLE  ALT=\"+if\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+cos+y+=+x ALIGN=MIDDLE  ALT=\"+cos+y+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "But, since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+cos+x ALIGN=MIDDLE  ALT=\"y+=+cos+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=not ALIGN=MIDDLE  ALT=\"not\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=one-to-one ALIGN=MIDDLE  ALT=\"one-to-one\"   DISABLED_style= \"max-width:90%; height:auto;\" >, its <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=domain ALIGN=MIDDLE  ALT=\"domain\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be restricted in order that y = cos^-1(x) is a function.\r<BR>\n" );
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document.write( "To get the graph of y = cos^-1x, start with a graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+cos+x ALIGN=MIDDLE  ALT=\"y+=+cos+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=plot_any_graph.solver>PLOT any graph</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Graphing function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29 ALIGN=MIDDLE  ALT=\"cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-4%2C+4%2C+-4%2C+4%2C+cos%28x%29+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-4%2C+4%2C+-4%2C+4%2C+cos%28x%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Restrict ALIGN=MIDDLE  ALT=\"Restrict\"   DISABLED_style= \"max-width:90%; height:auto;\" > the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=domain ALIGN=MIDDLE  ALT=\"domain\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the function to a one-to-one region - typically [<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >] is used for cos^ -1x. This leaves the range of the restricted function unchanged as [<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >].\r<BR>\n" );
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document.write( "Reflect the graph across the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x ALIGN=MIDDLE  ALT=\"y+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get the graph of y = cos^-1 x y = arccos x, the black curve at right on the graph above.\r<BR>\n" );
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document.write( "Notice that y = cos^-1x has domain [<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+1 ALIGN=MIDDLE  ALT=\"+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >] and range [<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >]. It is strictly decreasing on its entire domain. \r<BR>\n" );
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