document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=991271>Question 991271</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the domain and range of the quadratic equation f(x)=x^2-8x-9 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #611132 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52798)</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=611132\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Hello,<BR>\n" );
document.write( "your question is incorrect.\r<BR>\n" );
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document.write( "The correct question is:\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Find the domain and range of the quadratic <U>function</U> &nbsp;&nbsp;f(x)=x^2-8x-9.\r<BR>\n" );
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document.write( "An equation has no domain. &nbsp;The term &nbsp;\"domain\" is not defined for equations.  \r<BR>\n" );
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document.write( "It is defined for functions.\r<BR>\n" );
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document.write( "It is not applicable for equations. &nbsp;It is applicable for functions.\r<BR>\n" );
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document.write( "The same is for range. &nbsp;It is for functions, &nbsp;not for equations.\r<BR>\n" );
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document.write( "OK. &nbsp;Now I am ready to answer this question:\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Find the domain and range of the quadratic function &nbsp;f(x)=x^2-8x-9.\r<BR>\n" );
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document.write( "1) &nbsp;The domain of a quadratic function is the entire number line, &nbsp;i.e &nbsp;the set of all real numbers. \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is true for any quadratic function. &nbsp;Particularly, &nbsp;it is true for the given function.\r<BR>\n" );
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document.write( "2) &nbsp;The range of a quadratic function is stretched from its minimal value to the positive infinity, &nbsp;if the parabola is &nbsp;U-shaped &nbsp;(has positive coefficient at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;as it is in your case). \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If the parabola is bottom-up &nbsp;(the quadratic function has negative coefficient at &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >), &nbsp;then its range is stretched from its maximal value to the negative infinity.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;So, &nbsp;in your case we need to find the minimal value of the quadratic function. &nbsp;For a quadratic function &nbsp;f(x) = &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;the minimum is reached at &nbsp;x = -<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2F%282a%29 ALIGN=MIDDLE  ALT=\"b%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and is equal to f(-<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2F%282a%29%29 ALIGN=MIDDLE  ALT=\"%28b%2F%282a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >). &nbsp;In your case &nbsp;&nbsp;-<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2F%282a%29 ALIGN=MIDDLE  ALT=\"b%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2F2 ALIGN=MIDDLE  ALT=\"8%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 4 &nbsp;and &nbsp;f(4) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E2+-8%2A4+-+9 ALIGN=MIDDLE  ALT=\"4%5E2+-8%2A4+-+9\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 16 - 32 - 9 = -25.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Therefore, &nbsp;the range of your function is semi-infinite segment &nbsp;&nbsp;[-25, <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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