document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=138649>Question 138649</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find if the function is even, odd or neither.\r<BR>\n" );
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document.write( "f(x)=x^3+4x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #101175 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=101175\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Remember, if <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;\" > then the function is an even function. If <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;\" > then the function is an odd function.\r<BR>\n" );
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document.write( "First, let's see if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E3%2B4x ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3%2B4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an even function.\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%2B4x ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3%2B4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given 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=%28-x%29%5E3%2B4%28-x%29 ALIGN=MIDDLE  ALT=\"f%28-x%29=%28-x%29%5E3%2B4%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace each x with -x.\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%5E3-4x ALIGN=MIDDLE  ALT=\"f%28-x%29=-x%5E3-4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify. Note: only the terms with an <b>odd</b> exponent will change in sign.\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%3C%3Ef%28-x%29 ALIGN=MIDDLE  ALT=\"f%28x%29%3C%3Ef%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this shows us that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E3%2B4x ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3%2B4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <b>not</b> an even function.\r<BR>\n" );
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document.write( "Now, let's see if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E3%2B4x ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3%2B4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an odd function.\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%2B4x ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3%2B4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given function.\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%2B4x%29 ALIGN=MIDDLE  ALT=\"-f%28x%29=-%28x%5E3%2B4x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate the entire function by placing a negative outside the function.\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%5E3-4x ALIGN=MIDDLE  ALT=\"-f%28x%29=-x%5E3-4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute and simplify.\r<BR>\n" );
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document.write( "Since f(-x)=-f(x), this shows us that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E3%2B4x ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3%2B4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an odd function.\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E3%2B4x ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3%2B4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an odd function.<BR>\n" );
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