document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1104775>Question 1104775</A>: <I><SPAN STYLE=\"word-break: break-all;\"> fonction of composition  (Fog)(x)=1/x^2   and g(x)= 2x+1   <BR>\n" );
document.write( "find f(x)=?\r<BR>\n" );
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document.write( "thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #719474 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=719474\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> fog(x) = 1/x^2<BR>\n" );
document.write( "g(x) = 2x+1<BR>\n" );
document.write( "find f(x) .....\r<BR>\n" );
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document.write( "you start with g(x) and wind up with fog(x).\r<BR>\n" );
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document.write( "fog(x) means f(g(x)) which means f(2x+1).\r<BR>\n" );
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document.write( "therefore, you have f(2x+1) = 1/x^2\r<BR>\n" );
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document.write( "you are looking for something that happens to 2x+1 that makes the result equal to 1/x^2.\r<BR>\n" );
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document.write( "i played around with it for a while and then decided to figure out what happens to (2x+1)^2 and how i can make the result equal to x^2.\r<BR>\n" );
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document.write( "so i squared (2x+1) and got 4x^2 + 4x + 1.\r<BR>\n" );
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document.write( "i needed to get rid of the 4x, so i subtracted 2 * (2x+1) from it to get:\r<BR>\n" );
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document.write( "4x^2 + 4x + 1 - 2 * (2x+1) = 4x^2 + 4x + 1 -4x - 2 which simplifed to 4x^2 - 1.\r<BR>\n" );
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document.write( "i then added 1 to it to get 4x^2.\r<BR>\n" );
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document.write( "i then divided it by 4 to get x^2.\r<BR>\n" );
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document.write( "i wound up with 4 / ((2x+1)^2 - 2*(2x+1) + 1)\r<BR>\n" );
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document.write( "i then replaced 2x+1 with x and got 4 / (x^2 - 2x + 1).\r<BR>\n" );
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document.write( "that became f(x).\r<BR>\n" );
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document.write( "to confirm i did the right thing, i then went through the process of converting f(x) and g(x) to fog(x).\r<BR>\n" );
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document.write( "i started with f(x) = 4/(x^2 - 2x + 1) and g(x) = 2x + 1\r<BR>\n" );
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document.write( "fog(x) is equal to f(g(x)) which is equal to f(2x+1).\r<BR>\n" );
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document.write( "this means replace x with 2x + 1 in f(x).\r<BR>\n" );
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document.write( "since f(x) = 4 / (x^2 - 2x + 1), then replacing x with (2x+1) results in fog(x) = 4 / ((2x+1)^2 - 2*(2x+1) + 1).\r<BR>\n" );
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document.write( "expanding this, i got fog(x) = 4 / (4x^2 + 4x + 1 - 4x - 2 + 1)\r<BR>\n" );
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document.write( "simplifying this, i got fog(x) = 4 / 4x^2.\r<BR>\n" );
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document.write( "simplifying further, i got fog(x) = 1/x^2.\r<BR>\n" );
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document.write( "looks like i got it right.\r<BR>\n" );
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document.write( "i went further to graph both 1/x^2 and 4/((2x+1)^2 - 2*(2x+1) + 1).\r<BR>\n" );
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document.write( "the graphs are identical, indicating that the equations are equivalent.\r<BR>\n" );
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document.write( "my graphs are shown below.\r<BR>\n" );
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document.write( "the first graph shows 1/x^2.\r<BR>\n" );
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document.write( "the second graph shows 4 / ((2x+1)^2 - 2*(2x+1) + 1)\r<BR>\n" );
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document.write( "the third graph show them both together, resulting in the same graph, indicating the equations are equivalent to each other.\r<BR>\n" );
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document.write( "the little circles on the left tell you which equation is being graphed.<BR>\n" );
document.write( "if they're filled in with the color of the graph, that graph is showing.<BR>\n" );
document.write( "you can see when they're both being graphed together, their graph is identical, confirming the equations are equivalent to each other.\r<BR>\n" );
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document.write( "your solution is f(x) = 4 / (x^2 - 2x + 1)\r<BR>\n" );
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document.write( " </SPAN>\n" );
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