document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1136739>Question 1136739</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If f(x) = x/sqr(1+x^2), prove by mathematical induction for n>=2 that f(f(...(f (x))...)) = x/sqr(1+nx^2), if there are n number of letter f's on the LHS.\r<BR>\n" );
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document.write( "Please help! I don't understand what f of f(x) means. What exactly am I subbing in when I am trying to prove true for n=2 and assuming true for n=k?\r<BR>\n" );
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document.write( "Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #754539 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=754539\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> if you have f(x) = x / sqrt(1 + x^2), then:\r<BR>\n" );
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document.write( "f(f(x)) simply replaces x with f(x) in the original equation.\r<BR>\n" );
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document.write( "therefore:\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+%2F+sqrt%281+%2B+x%5E2%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+x+%2F+sqrt%281+%2B+x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the original equation, and ......\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28f%28x%29%29+=+%28x+%2F+sqrt%281+%2B+x%5E2%29%29%2Fsqrt%281+%2B+%28x+%2F+sqrt%281+%2B+x%5E2%29%29%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "this may be easier to see in my worksheet shown below.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/031501.JPG\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "in my worksheet, i carried it through to the end and got the final answer of  f(f(x)) = x / sqrt(1 + 2x^2)\r<BR>\n" );
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document.write( "if follows that, if you then do f(f(f(x)),the answer will be x / sqrt(1 + 3x^3), etc.\r<BR>\n" );
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document.write( "here's a reference on composite functions which you may already be familiar with.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.mathsisfun.com/sets/functions-composition.html\" target = \"_blank\">https://www.mathsisfun.com/sets/functions-composition.html</a>\r<BR>\n" );
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document.write( "here's a much simpler example of f(f(x)).\r<BR>\n" );
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document.write( "assume f(x) = x^2.\r<BR>\n" );
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document.write( "f(f(x)) would then replace x with x^2 and you would get f(f(x)) = (x^2)^2.\r<BR>\n" );
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