document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200539>Question 1200539</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The function f is defined f(x) = x^2 - 6x + 2  for 0 ≤ x ≤ 3<BR>\n" );
document.write( "a) state the range of f<BR>\n" );
document.write( "b) state the domain of f^-1 (x)<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #834753 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=834753\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "<font color=red>Answers:</font><BR>\n" );
document.write( "<ol type=\"a\"><li><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7%3C=+y+%3C=2 ALIGN=MIDDLE  ALT=\"-7%3C=+y+%3C=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which has the interval notation [-7, 2]</li><li><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7%3C=+x+%3C=2 ALIGN=MIDDLE  ALT=\"-7%3C=+x+%3C=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which has the interval notation [-7, 2]</li></ol>Explanation:\r<BR>\n" );
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document.write( "f(x) is defined only when 0 ≤ x ≤ 3\r<BR>\n" );
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document.write( "Refer to the 2nd graph the tutor Edwin McCravy had made.<BR>\n" );
document.write( "That graph shows (3,-7) is the lowest point on the parabola. \r<BR>\n" );
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document.write( "When completing the square, <BR>\n" );
document.write( "f(x) = x^2-6x+2<BR>\n" );
document.write( "becomes<BR>\n" );
document.write( "f(x) = (x-3)^2-7<BR>\n" );
document.write( "Use the FOIL rule to confirm this is the correct vertex form.\r<BR>\n" );
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document.write( "The highest point is (0,2)<BR>\n" );
document.write( "We can determine this by plugging x = 0 to get y = 2.\r<BR>\n" );
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document.write( "The range is the set of possible y outputs.<BR>\n" );
document.write( "y = -7 is the smallest output<BR>\n" );
document.write( "y = 2 is the largest output<BR>\n" );
document.write( "Therefore, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7%3C=+y+%3C=2 ALIGN=MIDDLE  ALT=\"-7%3C=+y+%3C=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents the range.<BR>\n" );
document.write( "y is between -7 and 2 including both endpoints.<BR>\n" );
document.write( "In interval notation, this is [-7, 2]<BR>\n" );
document.write( "The square brackets tell the reader to include the endpoints.\r<BR>\n" );
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document.write( "Recall the process of finding the inverse means we swap x and y, then solve for y.<BR>\n" );
document.write( "Because of this swap, the domain and range swap when going from the original function to the inverse.\r<BR>\n" );
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document.write( "Therefore,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{range of original} \leftrightarrow \text{domain of inverse}\"><BR>\n" );
document.write( "which means<BR>\n" );
document.write( "original has range <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7%3C=+y+%3C=2 ALIGN=MIDDLE  ALT=\"-7%3C=+y+%3C=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > leads to the inverse having the domain <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7%3C=+x+%3C=2 ALIGN=MIDDLE  ALT=\"-7%3C=+x+%3C=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. All we do is swap y for x.<BR>\n" );
document.write( "You can use the last graph Edwin McCravy had made as visual confirmation. \r<BR>\n" );
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document.write( "<a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a> and <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a> are two of many graphing tools you can use to verify the answer.<BR>\n" );
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