document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=243864>Question 243864</A>: <I><SPAN STYLE=\"word-break: break-all;\"> FOR THE INDICATED FUNCTIONS F AND G,FIND FUNCTIONS F0G AND G0F, FIND THE DOMAINS.<BR>\n" );
document.write( "F(X)=X^2;G(X)=X^3+2X+4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #178590 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=178590\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x) = x^2<BR>\n" );
document.write( "g(x) = x^3 + 2x + 4\r<BR>\n" );
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document.write( "to get f(g(x)), you start with f(x) = x^2\r<BR>\n" );
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document.write( "you replace x with g(x) to get:\r<BR>\n" );
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document.write( "f(g(x)) = (g(x))^2\r<BR>\n" );
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document.write( "you replace g(x) with x^3 + 2x + 4 to get:\r<BR>\n" );
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document.write( "f(g(x)) = (x^3 + 2x + 4)^2\r<BR>\n" );
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document.write( "you simplify to get:\r<BR>\n" );
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document.write( "f(g(x) = x^6 + 4x^4 + 8x^3 + 4x^2 + 16x + 16\r<BR>\n" );
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document.write( "you confirm your answer is good by taking any value of x and substituting i the original equation and the final equation to see if the equations are true.\r<BR>\n" );
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document.write( "I did with x = 2 and confirmed the answers were the same so I believe the answer is good.\r<BR>\n" );
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document.write( "to get g(f(x)):\r<BR>\n" );
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document.write( "you start with g(x) = x^3 + 2x + 4\r<BR>\n" );
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document.write( "you replace x with f(x) to get:\r<BR>\n" );
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document.write( "g(f(x)) = (f(x))^3 + 2*(f(x)) + 4\r<BR>\n" );
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document.write( "you replace f(x) with x^2 to get:\r<BR>\n" );
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document.write( "g(f(x)) = (x^2)^3 + 2*(x^2) + 4\r<BR>\n" );
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document.write( "you simplify to get:\r<BR>\n" );
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document.write( "g(f(x)) = x^6 + 2x^2 + 4\r<BR>\n" );
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document.write( "You confirm by replacing x with a random value and solving both the original equation and the final equation to see if they are the same.\r<BR>\n" );
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document.write( "I did with x = 3 and both equations came out the same so I believe the answer is good.\r<BR>\n" );
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document.write( "The domains are all real values of x since there is no division of 0 and there are no even roots of negative numbers (square root, fourth root, etc).   Those, if present, would restrict the domain.\r<BR>\n" );
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document.write( "Graph of x^2 looks like this:\r<BR>\n" );
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document.write( "Graph of x^3 + 2x + 4 looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-20%2C20%2C-20%2C20%2Cx%5E3%2B2x%2B4%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-20%2C20%2C-20%2C20%2Cx%5E3%2B2x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Graph of x^6 + 4x^4 + 8x^3 + 4x^2 + 16x + 16 looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-20%2C20%2C-100%2C100%2Cx%5E6+%2B+4x%5E4+%2B+8x%5E3+%2B+4x%5E2+%2B+16x+%2B+16%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Graph of x^6 + 2x^2 + 4 looks like this:\r<BR>\n" );
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document.write( "In the last 2 graphs, the scale of the y-axis was changed from +/- 20 to +/- 100 because they become very large.\r<BR>\n" );
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