document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=293389>Question 293389</A>: <I><SPAN STYLE=\"word-break: break-all;\"> can you tell me i have been trying to get the answer and I just can't get it . This two function with the works given f(x)= 1/x-2 G(x)=4/x.  Find (fog)(x) and  using the same function find (gof)(x). thank you in advance </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #211787 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=211787\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28x-2%29 ALIGN=MIDDLE  ALT=\"1%2F%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "g(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Fx ALIGN=MIDDLE  ALT=\"4%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Find (fog)(x)\r<BR>\n" );
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document.write( "Find (gof)(x)\r<BR>\n" );
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document.write( "First we'll do (fog)(x):\r<BR>\n" );
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document.write( "(fog)(x) = f(g(x))\r<BR>\n" );
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document.write( "g(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2Fx%29 ALIGN=MIDDLE  ALT=\"%284%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "f(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F%28x-2%29%29 ALIGN=MIDDLE  ALT=\"%281%2F%28x-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "f(g(x)) becomes f(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2Fx%29 ALIGN=MIDDLE  ALT=\"%284%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "You are replacing x with g(x) in f(x) which means you are replacing x with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Fx ALIGN=MIDDLE  ALT=\"4%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > in f(x).\r<BR>\n" );
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document.write( "Your equation f(x) becomes f(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2Fx%29 ALIGN=MIDDLE  ALT=\"%284%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "Since f(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28x-2%29 ALIGN=MIDDLE  ALT=\"1%2F%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you are replacing x with (4/x), then your equation becomes:\r<BR>\n" );
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document.write( "f(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2Fx%29 ALIGN=MIDDLE  ALT=\"%284%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28%284%2Fx%29-2%29 ALIGN=MIDDLE  ALT=\"1%2F%28%284%2Fx%29-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If you multiply numerator and denominator of this expression by x, then you get:\r<BR>\n" );
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document.write( "f(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2Fx%29 ALIGN=MIDDLE  ALT=\"%284%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2F+%28x%2A%28%284%2Fx%29+-+2%29%29 ALIGN=MIDDLE  ALT=\"x+%2F+%28x%2A%28%284%2Fx%29+-+2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes:\r<BR>\n" );
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document.write( "f(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2Fx%29 ALIGN=MIDDLE  ALT=\"%284%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2F+%284-2x%29 ALIGN=MIDDLE  ALT=\"x+%2F+%284-2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "To test this out, pick a number for x.\r<BR>\n" );
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document.write( "Let x = 5.\r<BR>\n" );
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document.write( "f(g(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Fx ALIGN=MIDDLE  ALT=\"4%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >)) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F%284-2x%29 ALIGN=MIDDLE  ALT=\"x%2F%284-2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F%284-8%29 ALIGN=MIDDLE  ALT=\"4%2F%284-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F%28-4%29 ALIGN=MIDDLE  ALT=\"4%2F%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = -1\r<BR>\n" );
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document.write( "Solve separately to confirm the answer is good.\r<BR>\n" );
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document.write( "Solve for g(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Fx ALIGN=MIDDLE  ALT=\"4%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get g(4) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F4 ALIGN=MIDDLE  ALT=\"4%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 1\r<BR>\n" );
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document.write( "You now have g(x) = 1\r<BR>\n" );
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document.write( "Solve for f(g(x)) = f(1) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28x-2%29 ALIGN=MIDDLE  ALT=\"1%2F%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%281-2%29 ALIGN=MIDDLE  ALT=\"1%2F%281-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28-1%29 ALIGN=MIDDLE  ALT=\"1%2F%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = -1\r<BR>\n" );
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document.write( "We get a match again, so the function f(g(x)) was translated correctly.\r<BR>\n" );
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document.write( "Next we'll do (gof)(x).\r<BR>\n" );
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document.write( "That's equivalent to g(f(x)).\r<BR>\n" );
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document.write( "f(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28x-2%29 ALIGN=MIDDLE  ALT=\"1%2F%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "g(x)= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Fx ALIGN=MIDDLE  ALT=\"4%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "g(f(x)) is therefore equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F%281%2F%28x-2%29%29 ALIGN=MIDDLE  ALT=\"4%2F%281%2F%28x-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Multiply numerator and denominator of this equation by (x-2) to get:\r<BR>\n" );
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document.write( "g(f(x)) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2A%28x-2%29%29%2F1 ALIGN=MIDDLE  ALT=\"%284%2A%28x-2%29%29%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 4x-8\r<BR>\n" );
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document.write( "To confirm we'll do g(x) separately, and then do f(g(x)) from the answer, and we'll do f(g(x)) using the combined formula to see if they match.\r<BR>\n" );
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document.write( "We'll choose x = 5 again.\r<BR>\n" );
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document.write( "g(f(x)) = 4x-8 = 4*5 - 8 = 20 - 8 = 12\r<BR>\n" );
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document.write( "Solve for f(x) separately to get f(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28x-2%29 ALIGN=MIDDLE  ALT=\"1%2F%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%285-2%29 ALIGN=MIDDLE  ALT=\"1%2F%285-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "solve for g(f(x)) = g(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >) to get g(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F%281%2F3%29 ALIGN=MIDDLE  ALT=\"4%2F%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A3 ALIGN=MIDDLE  ALT=\"4%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 12.\r<BR>\n" );
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document.write( "The answers match.\r<BR>\n" );
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document.write( "Those should be your answers:\r<BR>\n" );
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document.write( "f(g(x)) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2F+%284-2x%29 ALIGN=MIDDLE  ALT=\"x+%2F+%284-2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "g(f(x)) = 4x-8\r<BR>\n" );
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document.write( " </SPAN>\n" );
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