document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=814829>Question 814829</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me sketch the following function by determining the critical points as follows: f(x)=sqrt(x-4): the domain and intercepts </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #490560 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39620)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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=490560\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Not sure if it is called a critical point, but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=4 ALIGN=MIDDLE  ALT=\"x%3E=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a requirement in Real Numbers for the domain of f(x).  The far-left point is the one which intersects the x axis, at (4,0).  In other way of saying, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%284%29=0 ALIGN=MIDDLE  ALT=\"f%284%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  No other axis intercepts.\r<BR>\n" );
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document.write( "Hopefully, a better explanation:<BR>\n" );
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document.write( "DOMAIN:  What value are acceptable for x?<BR>\n" );
document.write( "Square Root function for real numbers must accept positive values for zero.  Negative values are not acceptable.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=sqrt%28x-4%29 ALIGN=MIDDLE  ALT=\"f%28x%29=sqrt%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > must have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-4%3E=0 ALIGN=MIDDLE  ALT=\"x-4%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=0%2B4 ALIGN=MIDDLE  ALT=\"x%3E=0%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=4 ALIGN=MIDDLE  ALT=\"x%3E=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "This means, the DOMAIN of f(x) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x%3E=4%29 ALIGN=MIDDLE  ALT=\"highlight%28x%3E=4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Checking for intercepts, at x=0, what is f(x)?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29=sqrt%280-4%29 ALIGN=MIDDLE  ALT=\"f%280%29=sqrt%280-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, NOT acceptable because 0 is not in the domain of f(x).  Recall, we just found that the domain of f(x) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=4 ALIGN=MIDDLE  ALT=\"x%3E=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C4 ALIGN=MIDDLE  ALT=\"0%3C4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so x cannot be 0.  This means, f(x) will not cross the y-axis.<BR>\n" );
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document.write( "What about f(x)=0?  What would x be for this?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=0=sqrt%28x-4%29 ALIGN=MIDDLE  ALT=\"f%28x%29=0=sqrt%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=sqrt%28x-4%29 ALIGN=MIDDLE  ALT=\"0=sqrt%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Square both sides,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=x-4 ALIGN=MIDDLE  ALT=\"0=x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is the point, (4,0) as the x-intercept.\r<BR>\n" );
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document.write( "Your will recognize the equation as half of a parabola with horizontal axis of symmetry:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-2%2C12%2C-2%2C7%2Csqrt%28x-4%29%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-2%2C12%2C-2%2C7%2Csqrt%28x-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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