document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1121642>Question 1121642</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you determine whether <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29+=+%28X%5E2%2B1%29%2F%28x%29 ALIGN=MIDDLE  ALT=\"+f%28x%29+=+%28X%5E2%2B1%29%2F%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an even, odd, or neither and what is its symmetry?\r<BR>\n" );
document.write( "\n" );
document.write( "If I recall correctly even tests are f(-x) = f(x), odd tests are f(-x) = -f(x) and even has a symmetry in respects to the y-axis and odd to the origin? But I don't know how to approach the above function. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #737601 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=737601\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "even tests are<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28-x%29+=+f%28x%29 ALIGN=MIDDLE  ALT=\"+f%28-x%29+=+f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
document.write( "\n" );
document.write( "given: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E2%2B1%29%2Fx+ ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%5E2%2B1%29%2Fx+\"   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-x%29=%28%28-x%29%5E2%2B1%29%2F%28-x%29 ALIGN=MIDDLE  ALT=\"f%28-x%29=%28%28-x%29%5E2%2B1%29%2F%28-x%29\"   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-x%29+=+-%28x%5E2%2B1%29%2Fx+ ALIGN=MIDDLE  ALT=\"f%28-x%29+=+-%28x%5E2%2B1%29%2Fx+\"   DISABLED_style= \"max-width:90%; height:auto;\" >=> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28-x%29+=+-f%28x%29 ALIGN=MIDDLE  ALT=\"+f%28-x%29+=+-f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "so,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29+%3C%3E+f%28x%29++ ALIGN=MIDDLE  ALT=\"f%28-x%29+%3C%3E+f%28x%29++\"   DISABLED_style= \"max-width:90%; height:auto;\" >=> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E2%2B1%29%2Fx ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%5E2%2B1%29%2Fx\"   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;\" > an even function\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "odd tests are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29+=+-f%28x%29 ALIGN=MIDDLE  ALT=\"f%28-x%29+=+-f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "since above is proven that f(-x) = -f(x), means f(x) = (x^2 + 1)/x is an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=odd ALIGN=MIDDLE  ALT=\"odd\"   DISABLED_style= \"max-width:90%; height:auto;\" > function\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the function is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+odd ALIGN=MIDDLE  ALT=\"+odd\"   DISABLED_style= \"max-width:90%; height:auto;\" >, it is  symmetric about the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=origin ALIGN=MIDDLE  ALT=\"origin\"   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=+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%28x%5E2%2B1%29%2Fx%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%28x%5E2%2B1%29%2Fx%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );