document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=627224>Question 627224</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what does the function exists and doesnot exist mean </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #394823 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=394823\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "A function exists for all values of the independent variable in the domain of the function.  A function does not exist for all values of the independent variable excluded from the domain of the function.\r<BR>\n" );
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document.write( "For example:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x)\ =\ \frac{1}{x\ -\ 3}\">  does not exist for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\ =\ 3\"> but does exist for all other real number values of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\">.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \psi\,:\,\mathbb{R}\ \rightarrow\ \mathbb{R},\ \psi(x)\ =\ \sqrt{x}\"> is defined, and therefore exists, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \forall\ x\ \in\ \mathbb{R}\,:\,x\ \geq\ 0\"> and is not defined, and therefore does not exist <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \forall\ x\ \in\ \mathbb{R}\,:\,x\ <\ 0\">.\r<BR>\n" );
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document.write( "On the other hand:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \psi\,:\,\mathbb{R}\ \rightarrow\ \mathbb{C},\ \psi(x)\ =\ \sqrt{x}\"> is defined, and therefore exists, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \forall\ x\ \in\ \mathbb{R}\">\r<BR>\n" );
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document.write( "Note: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \mathbb{R}\"> is the set of all real numbers and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \mathbb{C}\"> is the set of complex numbers.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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