document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186928>Question 1186928</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For each of the following statements, explain why it is true for all functions f and g or an example of specific functions f and g showing that it is false.\r<BR>\n" );
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document.write( "a. All elements in the domain of g must be in the domain of f o g (fog)<BR>\n" );
document.write( "b. All elements in the range of f must be in the range of f o g </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849343 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52858)</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=849343\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "For each of the following statements, explain why it is true for all functions f and g or an example of specific functions f and g showing that it is false.\r<BR>\n" );
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document.write( "a. All elements in the domain of g must be in the domain of f o g (fog)<BR>\n" );
document.write( "b. All elements in the range of f must be in the range of f o g<BR>\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~\r<BR>\n" );
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document.write( "In the post by &nbsp;@CPhill &nbsp;(Artificial &nbsp;Intelligence &nbsp;?), &nbsp;&nbsp;the answer &nbsp;\" True \" &nbsp;in part &nbsp;(a) &nbsp;is a &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%28FATAL%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%28FATAL%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%28ERROR%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%28ERROR%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Counter-example:  &nbsp;&nbsp;g(x) = x; &nbsp;&nbsp;f(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx ALIGN=MIDDLE  ALT=\"1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Then &nbsp;x = 0  &nbsp;is in the domain of &nbsp;g, &nbsp;&nbsp;but is &nbsp;NOT &nbsp;in the domain of &nbsp;&nbsp;(fog)(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx ALIGN=MIDDLE  ALT=\"1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "By the way, it means that all the argumentation of @CPhill in part &nbsp;(a) &nbsp;is invalid,<BR>\n" );
document.write( "although it produces quite solid impression.\r<BR>\n" );
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document.write( "A stern examiner would give a solid &nbsp;\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%282%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \" &nbsp;to such an &nbsp;\" artificial intelligence \" &nbsp;student.\r<BR>\n" );
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document.write( "But remember, &nbsp;that an &nbsp;\" artificial intelligence\" &nbsp;claims not to be &nbsp;\" a student \", \r<BR>\n" );
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document.write( "but to be a &nbsp;\" SUPERIOR &nbsp;MIND \", &nbsp;who does not make mistakes, &nbsp;knows everything \r<BR>\n" );
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document.write( "and leads everybody around to a bright future &nbsp;(&nbsp;!&nbsp;)\r<BR>\n" );
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document.write( "What I want to say is that still there are plenty of the room to improve your skills\r<BR>\n" );
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document.write( "as an \"artificial intelligence\" before you become a true \"Artificial Intelligence\".\r<BR>\n" );
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document.write( " </SPAN>\n" );
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