document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186585>Question 1186585</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The function f is such that f(x) = 3x - 2 for x >= 0.<BR>\n" );
document.write( "The function g is such that g(x) = 2x^2 - 8 for x <= k, where k is a constant.\r<BR>\n" );
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document.write( "(a) Find the greatest value of k for which the composite function fg can be formed.\r<BR>\n" );
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document.write( "(b) For the case where  k = -3\r<BR>\n" );
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document.write( "    (i) find the range of fg<BR>\n" );
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document.write( "    (ii) find (fg)^-1(x) and state the domain and the range of (fg)^-1. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #817507 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=817507\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given that the (restricted) domain of f(x) = 3x - 2 is [0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >), we need  to choose the values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+2x%5E2+-+8+%3E=+0 ALIGN=MIDDLE  ALT=\"g%28x%29+=+2x%5E2+-+8+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the composition f o g to be possible.\r<BR>\n" );
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document.write( "(a)  Hence, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+-+8+%3E=+0 ALIGN=MIDDLE  ALT=\"2x%5E2+-+8+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%3E=+4 ALIGN=MIDDLE  ALT=\"x%5E2+%3E=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or x ∈ (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity ALIGN=MIDDLE  ALT=\"-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >, -2] U [2, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >).  By virtue of continuity, the greatest value of k for which the composite function f o g can be formed is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28-2%29 ALIGN=MIDDLE  ALT=\"red%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "(b)  (i)  Now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f+o+g%29%28x%29+=+f%28g%28x%29%29+=+3%282x%5E2+-+8%29-2+=+6x%5E2++-26 ALIGN=MIDDLE  ALT=\"%28f+o+g%29%28x%29+=+f%28g%28x%29%29+=+3%282x%5E2+-+8%29-2+=+6x%5E2++-26\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Since the domain of f o g is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity ALIGN=MIDDLE  ALT=\"-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >, -3], the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2++-26 ALIGN=MIDDLE  ALT=\"6x%5E2++-26\"   DISABLED_style= \"max-width:90%; height:auto;\" > maps (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity ALIGN=MIDDLE  ALT=\"-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >, -3] onto the interval [28, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >).  <BR>\n" );
document.write( "This is the range of (f o g)(x).\r<BR>\n" );
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document.write( "(ii)  To find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f+o+g%29%5E%28-1%29%28x%29 ALIGN=MIDDLE  ALT=\"%28f+o+g%29%5E%28-1%29%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=++6x%5E2++-26 ALIGN=MIDDLE  ALT=\"y+=++6x%5E2++-26\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+%28y%2B26%29%2F6 ALIGN=MIDDLE  ALT=\"x%5E2+=+%28y%2B26%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >  ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-sqrt%28%28y%2B26%29%2F6%29 ALIGN=MIDDLE  ALT=\"x+=+-sqrt%28%28y%2B26%29%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  (Choose the negative part since this is an element of (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity ALIGN=MIDDLE  ALT=\"-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >, -3].)\r<BR>\n" );
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document.write( "===>  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-sqrt%28%28x%2B26%29%2F6%29 ALIGN=MIDDLE  ALT=\"y+=+-sqrt%28%28x%2B26%29%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > after interchanging the places of x and y.\r<BR>\n" );
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document.write( "===>  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f+o+g%29%5E%28-1%29%28x%29+=+-sqrt%28%28x%2B26%29%2F6%29 ALIGN=MIDDLE  ALT=\"%28f+o+g%29%5E%28-1%29%28x%29+=+-sqrt%28%28x%2B26%29%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Its domain is the range of f o g, namely [28, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >), while its range is the domain of f o g, namely (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity ALIGN=MIDDLE  ALT=\"-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >, -3].\r<BR>\n" );
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document.write( " </SPAN>\n" );
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