document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210389>Question 1210389</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The function f(x) = (1\x) is decreasing on its domain ( true or false)\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852170 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52754)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852170\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;First of all, &nbsp;I read this function as &nbsp;&nbsp;f(x) = 1/x, \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;following to the standard rules commonly accepted in &nbsp;Math.\r<BR>\n" );
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document.write( "The domain is the set of all real numbers except of 0 (zero).\r<BR>\n" );
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document.write( "Function &nbsp;&nbsp;f(x) = 1/x &nbsp;&nbsp;has two branches: one over the set of positive &nbsp;'x',<BR>\n" );
document.write( "another over the set of negative &nbsp;'x'.\r<BR>\n" );
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document.write( "For each branch separately, &nbsp;function &nbsp;&nbsp;f(x) = 1/x &nbsp;&nbsp;is monotonic in each  sub-domain.\r<BR>\n" );
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document.write( "Or, more accurately, &nbsp;function &nbsp;&nbsp;f(x) = 1/x &nbsp;&nbsp;is monotonically decreases locally in each sib-domain.\r<BR>\n" );
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document.write( "To prove it, &nbsp;take the derivative: &nbsp;it is  &nbsp;&nbsp;- <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx%5E2 ALIGN=MIDDLE  ALT=\"1%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The denominator &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;is always positive for all &nbsp;'x' &nbsp;of the domain; &nbsp;&nbsp;so, the derivative is always negative.\r<BR>\n" );
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document.write( "You also can convince yourself by making a plot of this function.\r<BR>\n" );
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document.write( "For it, &nbsp;go to website https:\\www.desmos.com/calculator/\r<BR>\n" );
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document.write( "Print the formula for the function  &nbsp;y = 1/x.\r<BR>\n" );
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document.write( "You will get the plot instantly.\r<BR>\n" );
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document.write( "The plot clearly shows that the function monotonically decreases as &nbsp;'x' &nbsp;increases in the domain.\r<BR>\n" );
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document.write( "So, &nbsp;locally for each branch, &nbsp;the function &nbsp;&nbsp;f(x) = 1/x &nbsp;&nbsp;monotonically decreases.<BR>\n" );
document.write( "You have this answer proved formally and demonstrated / illustrated visually.\r<BR>\n" );
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document.write( "But we can not say that function &nbsp;&nbsp;f(x) = 1/x &nbsp;&nbsp;is globally decreasing: &nbsp;&nbsp;when &nbsp;'x' &nbsp;&nbsp;moves <BR>\n" );
document.write( "from negative values to positive values, &nbsp;the function &nbsp;1/x &nbsp;jumps from negative values to positive,<BR>\n" );
document.write( "breaking monotonicity.\r<BR>\n" );
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document.write( "So, &nbsp;locally this function is monotonically decreasing, &nbsp;but globally it is not monotonic.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This problem is an elementary  &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%28TRAP%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%28TRAP%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "As I explained in my post above, &nbsp;locally and piece-wisely, &nbsp;the function is monotonically decreasing.\r<BR>\n" );
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document.write( "But globally, &nbsp;it is &nbsp;NOT &nbsp;monotonic.\r<BR>\n" );
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document.write( "It is &nbsp;OBVIOUS: &nbsp;compare these values of the function\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "    x     -2     -1     1     2   \r\n" );
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document.write( "    1/x   -0.5   -1     1     0.5\r\n" );
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document.write( "          decrease      decrease\r\n" );
document.write( "                 increase\r\n" );
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document.write( "They show non-monotonic behavior.\r<BR>\n" );
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document.write( "Tutor &nbsp;@math_tutor2000 &nbsp;fell into this trap.\r<BR>\n" );
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