document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=87302>Question 87302</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the symmetry (relative to the x-axis, y-axis, and/or origin) of the graph of the function. No need for graph. f(x)= x^3 + x  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #63241 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=63241\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Remember, a function is even when this equality is true:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=f%28-x%29 ALIGN=MIDDLE  ALT=\"f%28x%29=f%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and a function is odd when this equality is true:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-f%28x%29=f%28-x%29 ALIGN=MIDDLE  ALT=\"-f%28x%29=f%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Lets see if the function is even:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E3%2Bx ALIGN=MIDDLE  ALT=\"y=x%5E3%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "Here's <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E3%2Bx ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" > This is the given equation\r<BR>\n" );
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document.write( "Here's <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29 ALIGN=MIDDLE  ALT=\"f%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29=%28-x%29%5E3%2B%28-x%29 ALIGN=MIDDLE  ALT=\"f%28-x%29=%28-x%29%5E3%2B%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace x with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x ALIGN=MIDDLE  ALT=\"-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29=-x%5E3-x ALIGN=MIDDLE  ALT=\"f%28-x%29=-x%5E3-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2Bx ALIGN=MIDDLE  ALT=\"x%5E3%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not equal <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E3-x ALIGN=MIDDLE  ALT=\"-x%5E3-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=f%28-x%29 ALIGN=MIDDLE  ALT=\"f%28x%29=f%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <b>not</b> true.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > does <b>not</b> equal <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29 ALIGN=MIDDLE  ALT=\"f%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E3%2Bx ALIGN=MIDDLE  ALT=\"y=x%5E3%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <b>not</b> an even function.\r<BR>\n" );
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document.write( "Now lets see if the function is odd:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E3%2Bx ALIGN=MIDDLE  ALT=\"y=x%5E3%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "Here's <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29 ALIGN=MIDDLE  ALT=\"f%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29=-x%5E3-x ALIGN=MIDDLE  ALT=\"f%28-x%29=-x%5E3-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Remember we solved for this previously\r<BR>\n" );
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document.write( "Here's <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-f%28x%29 ALIGN=MIDDLE  ALT=\"-f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-f%28x%29=-%28x%5E3%2Bx%29 ALIGN=MIDDLE  ALT=\"-f%28x%29=-%28x%5E3%2Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate the whole function\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29=-x%5E3-x ALIGN=MIDDLE  ALT=\"f%28-x%29=-x%5E3-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute the negative and simplify\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E3-x ALIGN=MIDDLE  ALT=\"-x%5E3-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E3-x ALIGN=MIDDLE  ALT=\"-x%5E3-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-f%28x%29=f%28-x%29 ALIGN=MIDDLE  ALT=\"-f%28x%29=f%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is true.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-f%28x%29 ALIGN=MIDDLE  ALT=\"-f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29 ALIGN=MIDDLE  ALT=\"f%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E3%2Bx ALIGN=MIDDLE  ALT=\"y=x%5E3%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an odd function. This means the equation has symmetry with respect to the origin.\r<BR>\n" );
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document.write( "When we graph the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E3%2Bx ALIGN=MIDDLE  ALT=\"y=x%5E3%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can see if the equation has any symmetry:\r<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+-10%2C+10%2C+-10%2C+10%2C+x%5E3%2Bx%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E3%2Bx%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and we can clearly see that the equation is symmetrical with respect to the origin\r<BR>\n" );
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