document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165235>Question 1165235</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that the function f is defined for all real numbers as follows.\r<BR>\n" );
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document.write( "f(x)= -x^(2)+10  if -4<=x<4\r<BR>\n" );
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document.write( "      -5-x       if x>=4\r<BR>\n" );
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document.write( "Graph the function f. Then determine whether or not the function is continuous. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852237 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52867)</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=852237\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Suppose that the function f is defined for all real numbers as follows.<BR>\n" );
document.write( "f(x)= -x^(2)+10 if -4<=x<4<BR>\n" );
document.write( "-5-x if x>=4<BR>\n" );
document.write( "Graph the function f. Then determine whether or not the function is continuous.<BR>\n" );
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document.write( "In each of the two domains, the function is defined as a polynomial.\r\n" );
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document.write( "Therefore, inside each domain the function is continuous (since a polynomial is \r\n" );
document.write( "a continuous function, as you should know from Calculus).\r\n" );
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document.write( "Hence, to be a continuous function over the entire domain, the necessary and sufficient condition \r\n" );
document.write( "for the given function is to have the limit at x= 4 from the left to be equal to its value at x = 4.\r\n" );
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document.write( "The limit at x = 4 from the left side is  -x^2 + 10 at x ---> 4,  i.e. -4^2+10 = -16+10 = -6.\r\n" );
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document.write( "The value of the function at  x = 4  is  -5 - 4 = -9.\r\n" );
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document.write( "-9 =/= -6, so we conclude that the given function is not continuous at x = 4.\r\n" );
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document.write( "At this point, we complete our reasoning and answered the question.\r<BR>\n" );
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document.write( "Notice that the problem's formulation has a deficiency.\r<BR>\n" );
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document.write( "Indeed, it says that \"the function f is defined for all real numbers as follows . . . \",<BR>\n" );
document.write( "but in reality it is defined only on the union of two intervals [-4,4) and [4,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "This union is the whole interval [<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >), but it is not the set of all real numbers,<BR>\n" );
document.write( "as the problem proclaims. Thus, the problems wording is not perfect and is not accurate.\r<BR>\n" );
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document.write( "By such signs, every professional Math reader always can unmistakably recognize <BR>\n" );
document.write( "\"problems\" that were written by an unprofessional \"in a garage on the knee.\"\r<BR>\n" );
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document.write( "By the way, the first three words in the problem \"Suppose that the\" are excessive. <BR>\n" );
document.write( "They are not necessary and can be omitted. Then the problem's appearance will be better.\r<BR>\n" );
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document.write( "It is a rule of Math writing: remove everything that does not matter.\r<BR>\n" );
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document.write( "In Math, following this rule is expressing of respect to a reader.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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