document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=287718>Question 287718</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine algebraically if the function f(x)= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x-x%5E2sinx+ ALIGN=MIDDLE  ALT=\"+x-x%5E2sinx+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is odd, even, or neither. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #208584 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=208584\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The function is even if:\r<BR>\n" );
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document.write( "f(x) = f(-x)\r<BR>\n" );
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document.write( "The function is odd if:\r<BR>\n" );
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document.write( "-f(x) = f(-x)\r<BR>\n" );
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document.write( "Here's a reference:\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://en.wikipedia.org/wiki/Even_and_odd_functions\" target = \"_blank\">http://en.wikipedia.org/wiki/Even_and_odd_functions</a>\r<BR>\n" );
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document.write( "Here's a graph of your function:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-30%2C30%2C-500%2C500%2Cx-x%5E2%2Asin%28x%29%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-30%2C30%2C-500%2C500%2Cx-x%5E2%2Asin%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "It's not even as far as I can tell.\r<BR>\n" );
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document.write( "It does look odd.\r<BR>\n" );
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document.write( "Only way to tell for sure is to give it the test.\r<BR>\n" );
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document.write( "First the even test.\r<BR>\n" );
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document.write( "Function is even if f(-x) = f(x)\r<BR>\n" );
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document.write( "f(x) = x - x^2 * sin(x)\r<BR>\n" );
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document.write( "f(-x) = (-x) - (-x)^2 * sin(-x) which becomes:\r<BR>\n" );
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document.write( "f(-x) = (-x) - (x^2 * (-sin(x))) which becomes:\r<BR>\n" );
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document.write( "f(-x) = -x + (x^2*sin(x))\r<BR>\n" );
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document.write( "Function is not even.\r<BR>\n" );
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document.write( "Next the odd test.\r<BR>\n" );
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document.write( "Function is odd if:\r<BR>\n" );
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document.write( "f(-x) = -f(x)\r<BR>\n" );
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document.write( "f(x) = x - x^2 * sin(x)\r<BR>\n" );
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document.write( "-f(x) = -(x - x^2 * sin(x)) which becomes:\r<BR>\n" );
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document.write( "-f(x) = -x + (x^2 * sin(x))\r<BR>\n" );
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document.write( "f(-x) from above became:\r<BR>\n" );
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document.write( "f(-x) = -x + (x^2 * sin(x)).\r<BR>\n" );
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document.write( "Looks like the function is odd.\r<BR>\n" );
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document.write( "We can test this out with numbers.\r<BR>\n" );
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document.write( "We know that the sine of 60 degrees = .866025404\r<BR>\n" );
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document.write( "60 degrees * pi/180 = 1.047197551 radians.\r<BR>\n" );
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document.write( "sine of 1.047197551 radians = .866025404\r<BR>\n" );
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document.write( "Since the sines are the same, we know we did the conversion from degrees to radians correctly.\r<BR>\n" );
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document.write( "When you graph the sine of a number, the x is assumed to be in radians.  \r<BR>\n" );
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document.write( "That's why this conversion was necessary.\r<BR>\n" );
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document.write( "We will now take f(x) = x - x^2 * sin(x) and solve for f(1.047197551)\r<BR>\n" );
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document.write( "f(1.047197551) = 1.047197551 - (1.047197551)^2 * sin(1.047197551).\r<BR>\n" );
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document.write( "That becomes f(1.047197551) = .097494425\r<BR>\n" );
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document.write( "Now we want to find f(-x) when x = 1.047197551.\r<BR>\n" );
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document.write( "f(-1.047197551) = (-1.047197551) - (-1.047197551)^2 * sin(-1.047197551).\r<BR>\n" );
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document.write( "That becomes f(-1.047197551) = -.097494425\r<BR>\n" );
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document.write( "f(x) = .097494425<BR>\n" );
document.write( "-f(x) = -.097494425<BR>\n" );
document.write( "f(-x) = -.097494425\r<BR>\n" );
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document.write( "-f(x) = f(-x) confirming that the function is odd.\r<BR>\n" );
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document.write( "When you look at the graph, you will see that it looks symmetric except that the signs are reversed.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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