document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=110971>Question 110971</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0321227573/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> explain why a polynomial function of even degree cannot have an inverse.\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #80971 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=80971\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> first recall the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=definition ALIGN=MIDDLE  ALT=\"definition\"   DISABLED_style= \"max-width:90%; height:auto;\" > of inverse: \r<BR>\n" );
document.write( "\n" );
document.write( "A function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" > with domain <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is said to be <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;\" > if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NO ALIGN=MIDDLE  ALT=\"NO\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "distinct points in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > have same <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=image ALIGN=MIDDLE  ALT=\"image\"   DISABLED_style= \"max-width:90%; height:auto;\" > under <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >, that is:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x1%29+ ALIGN=MIDDLE  ALT=\"f%28x1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=is_+not_+equal ALIGN=MIDDLE  ALT=\"is_+not_+equal\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x2%29 ALIGN=MIDDLE  ALT=\"f%28x2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > whenever <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x1 ALIGN=MIDDLE  ALT=\"x1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=is_+not_+%0D%0A%0D%0Aequal ALIGN=MIDDLE  ALT=\"is_+not_+%0D%0A%0D%0Aequal\"   DISABLED_style= \"max-width:90%; height:auto;\" >  to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x2 ALIGN=MIDDLE  ALT=\"x2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x1 ALIGN=MIDDLE  ALT=\"x1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x2 ALIGN=MIDDLE  ALT=\"x2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is element of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( " Why a polynomial function of even degree cannot have an inverse? \r<BR>\n" );
document.write( "\n" );
document.write( "Simply, because a polynomial function of even degree 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;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "<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;\" > function.\r<BR>\n" );
document.write( "\n" );
document.write( "Each value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > squared, raised to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4th ALIGN=MIDDLE  ALT=\"4th\"   DISABLED_style= \"max-width:90%; height:auto;\" > degree, or higher even \r<BR>\n" );
document.write( "\n" );
document.write( "degree, will be that same value; for example, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%5E2=+4 ALIGN=MIDDLE  ALT=\"-2%5E2=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and also <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E2+=+%0D%0A%0D%0A4 ALIGN=MIDDLE  ALT=\"2%5E2+=+%0D%0A%0D%0A4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This means that two distinct points in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > have same <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=image ALIGN=MIDDLE  ALT=\"image\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "under <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "example:\r<BR>\n" );
document.write( "\n" );
document.write( "let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+1+-+x%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29+=+1+-+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "let domain <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > be equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   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;\" >,<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=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-3%29 ALIGN=MIDDLE  ALT=\"f%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-1%29 ALIGN=MIDDLE  ALT=\"f%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29 ALIGN=MIDDLE  ALT=\"f%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29 ALIGN=MIDDLE  ALT=\"f%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%283%29 ALIGN=MIDDLE  ALT=\"f%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-3%29=+1+-+%28-3%29%5E2=+1+-+9+=+-8 ALIGN=MIDDLE  ALT=\"f%28-3%29=+1+-+%28-3%29%5E2=+1+-+9+=+-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-1%29=+1+-+%28-1%29%5E2=+1+-+1+=+0 ALIGN=MIDDLE  ALT=\"f%28-1%29=+1+-+%28-1%29%5E2=+1+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29=+1+-+%280%29%5E2=+1+-+0+=+1 ALIGN=MIDDLE  ALT=\"f%280%29=+1+-+%280%29%5E2=+1+-+0+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29=+1+-+%281%29%5E2=+1+-+1+=+0 ALIGN=MIDDLE  ALT=\"f%281%29=+1+-+%281%29%5E2=+1+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%283%29=+1+-+%283%29%5E2=+1+-+9+=+-8 ALIGN=MIDDLE  ALT=\"f%283%29=+1+-+%283%29%5E2=+1+-+9+=+-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "as you can see,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-3%29 ALIGN=MIDDLE  ALT=\"f%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%283%29 ALIGN=MIDDLE  ALT=\"f%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > have same image under <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "also <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-1%29 ALIGN=MIDDLE  ALT=\"f%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29 ALIGN=MIDDLE  ALT=\"f%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > have same image under <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );