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 #852240 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852240\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "This is the graph of the piecewise function.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-15%2C15%2C-15%2C15%2C%0D%0Agraph%28400%2C400%2C-15%2C15%2C-15%2C15%2C%28%28sqrt%2815.80063-x%5E2%29%29%2F%28sqrt%2815.80063-x%5E2%29%29%29%2A%28-x%5E2%2B10%29%29%2C%0D%0Ared%28%0D%0Acircle%28-4%2C-6%2C0.29%29%2C%0D%0Acircle%28-4%2C-6%2C0.31%29%2C%0D%0Acircle%28-4%2C-6%2C0.33%29%2C%0D%0Acircle%28-4%2C-6%2C0.35%29%2C%0D%0Acircle%28-4%2C-6%2C0.37%29%2C%0D%0Acircle%28-4%2C-6%2C0.39%29%2C%0D%0Acircle%28-4%2C-6%2C0.41%29%2C%0D%0Acircle%28-4%2C-6%2C0.43%29%2C%0D%0Acircle%28-4%2C-6%2C0.45%29%2C%0D%0Acircle%28-4%2C-6%2C0.47%29%2C%0D%0Acircle%28-4%2C-6%2C0.49%29%2C%0D%0A%0D%0Acircle%284%2C-6%2C0.45%29%2C%0D%0Acircle%284%2C-6%2C0.47%29%2C%0D%0Acircle%284%2C-6%2C0.49%29%0D%0A%29%2C%0D%0A%0D%0Ablue%28%0D%0Aline%284%2C-9%2C20%2C-25%29%2C%0D%0Acircle%284%2C-9%2C0.25%29%2C%0D%0Acircle%284%2C-9%2C0.27%29%2C%0D%0Acircle%284%2C-9%2C0.29%29%2C%0D%0Acircle%284%2C-9%2C0.31%29%2C%0D%0Acircle%284%2C-9%2C0.33%29%2C%0D%0Acircle%284%2C-9%2C0.35%29%2C%0D%0Acircle%284%2C-9%2C0.37%29%2C%0D%0Acircle%284%2C-9%2C0.39%29%2C%0D%0Acircle%284%2C-9%2C0.41%29%2C%0D%0Acircle%284%2C-9%2C0.43%29%2C%0D%0Acircle%284%2C-9%2C0.45%29%2C%0D%0Acircle%284%2C-9%2C0.47%29%2C%0D%0Acircle%284%2C-9%2C0.49%29%0D%0A%29%2C%0D%0A%0D%0Ared%28locate%28-7%2C-7%2C%22%28-4%2C-6%29%22%29%29%2C%0D%0Ared%28locate%284.5%2C-5%2C%22%284%2C-6%29%22%29%29%2C%0D%0Ablue%28locate%284.5%2C-8%2C%22%284%2C-9%29%22%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The red parabola is due to the piece y = -x^2+10, when graphed on the interval -4 <= x < 4.<BR>\n" );
document.write( "Graph y = -x^2+10 as normal, then erase the portions when x < -4 or x > 4.<BR>\n" );
document.write( "The vertex is at (0,10). <BR>\n" );
document.write( "Two other points on this parabola are (-1,9) and (1,9).\r<BR>\n" );
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document.write( "Two graphing tools I recommend are <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a> and <a  rel=nofollow HREF=\"https://www.geogebra.org/graphing?lang=en\">GeoGebra</a><BR>\n" );
document.write( "Or you can use something like a TI83.<BR>\n" );
document.write( "Another alternative is to use graph paper to do everything by hand.\r<BR>\n" );
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document.write( "The blue line is the piece y = -5-x on the interval x >= 4.<BR>\n" );
document.write( "Two points on this blue line are (4,-9) and (5,-10).\r<BR>\n" );
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document.write( "There is a red closed endpoint at (-4,-6)<BR>\n" );
document.write( "There is an open hole at (4,-6) marked in red.<BR>\n" );
document.write( "There is a blue closed filled in endpoint at (4,-9)<BR>\n" );
document.write( "Another way of saying \"closed endpoint\" could be \"filled in endpoint\".\r<BR>\n" );
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document.write( "The open endpoint is due to the lack of \"or equal to\" in the inequality sign. <BR>\n" );
document.write( "This is why we exclude x = 4 from the red parabola.<BR>\n" );
document.write( "In contrast every other endpoint is included because there is an \"or equal to\".\r<BR>\n" );
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document.write( "We can clearly see the graph is <u>not</u> continuous. <BR>\n" );
document.write( "There's a jump at x = 4 when going from the red curve to the blue line.\r<BR>\n" );
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document.write( "To determine this algebraically, without needing a graph, plug x = 4 into both pieces to see what results.<BR>\n" );
document.write( "I'll label the pieces g(x) and h(x).<BR>\n" );
document.write( "g(x) = -x^2+10<BR>\n" );
document.write( "g(4) = -4^2+10<BR>\n" );
document.write( "g(4) = -16+10<BR>\n" );
document.write( "g(4) = -6<BR>\n" );
document.write( "This leads to the location (4,-6) which is the red open hole on the graph.\r<BR>\n" );
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document.write( "And,<BR>\n" );
document.write( "h(x) = -5-x<BR>\n" );
document.write( "h(4) = -5-4<BR>\n" );
document.write( "h(4) = -9<BR>\n" );
document.write( "This leads to the blue filled in endpoint at (4,-9)<BR>\n" );
document.write( "The results -6 and -9 do not agree on the same number, so this is a non-graph confirmation that the piecewise function is <u>not</u> continuous.<BR>\n" );
document.write( "To be continuous, we would need g(4) = h(4) to be the case. \r<BR>\n" );
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document.write( "Similar practice problems are at the following links<BR>\n" );
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