document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1052272>Question 1052272</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If f(x)=5/x and g(x)=2/2x+1 , find the functions (a,b) and their domains.\r<BR>\n" );
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document.write( "a.) (f o g)(x) I did 5/1/2/2x+1 and I multiplied 2 and 1 and multiplied 5 and 2x+1 to get 10x+5/2 . I then factored out the 5 in the numerator to get 5(2x+1) (Not positive if I need to factor the numerator or not) and then the denominator stayed at 2. I think the correct answer to this is 5(2x+1)/2 . And the Domain= (-infinity, 1/2)u(1/2, +infinity). (Not sure if this Domain is correct, nor am I sure how I got it).\r<BR>\n" );
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document.write( "b.) (g o f)(x) I am not sure how to solve this one. I am stuck at 2/2(5/x)+1. I am not sure how to get the fraction out of the denominator. \r<BR>\n" );
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document.write( "Anything helps, thank you.<BR>\n" );
document.write( "-Garrett </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #667672 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=667672\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Part A\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+5%2Fx ALIGN=MIDDLE  ALT=\"f%28x%29+=+5%2Fx\"   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=f%28g%28x%29%29+=+5%2F%28g%28x%29%29 ALIGN=MIDDLE  ALT=\"f%28g%28x%29%29+=+5%2F%28g%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace EVERY copy of x with g(x)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28g%28x%29%29+=+5%2F%282%2F%282x%2B1%29%29 ALIGN=MIDDLE  ALT=\"f%28g%28x%29%29+=+5%2F%282%2F%282x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Focus only on the right side. Replace g(x) with it's equivalent expression. Now simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28g%28x%29%29+=+%285%2F1%29%2F%282%2F%282x%2B1%29%29 ALIGN=MIDDLE  ALT=\"f%28g%28x%29%29+=+%285%2F1%29%2F%282%2F%282x%2B1%29%29\"   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=f%28g%28x%29%29+=+%285%282x%2B1%29%29%2F%281%2A2%29 ALIGN=MIDDLE  ALT=\"f%28g%28x%29%29+=+%285%282x%2B1%29%29%2F%281%2A2%29\"   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=f%28g%28x%29%29+=+%2810x%2B5%29%2F%282%29 ALIGN=MIDDLE  ALT=\"f%28g%28x%29%29+=+%2810x%2B5%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since <font color=blue>(f o g)(x)</font> is the same as <font color=blue>f(g(x))</font>, this means that <font color=blue>(f o g)(x)</font> is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810x%2B5%29%2F%282%29 ALIGN=MIDDLE  ALT=\"%2810x%2B5%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The simplified form of the equation above leads to no domain issues. However, if you consider the original function before you simplify, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-1%2F2 ALIGN=MIDDLE  ALT=\"x+=+-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > leads to a division by zero error. <BR>\n" );
document.write( "So we must kick that out of the domain.\r<BR>\n" );
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document.write( "So the domain of (f o g)(x) is <font color=red>(-infinity, -1/2) U (-1/2, infinity)</font>. Any other number works in the domain. \r<BR>\n" );
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document.write( "====================================================================\r<BR>\n" );
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document.write( "Part B\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+2%2F%282x%2B1%29 ALIGN=MIDDLE  ALT=\"g%28x%29+=+2%2F%282x%2B1%29\"   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=g%28f%28x%29%29+=+2%2F%282%2Af%28x%29%2B1%29 ALIGN=MIDDLE  ALT=\"g%28f%28x%29%29+=+2%2F%282%2Af%28x%29%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace EVERY copy of x with f(x)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28f%28x%29%29+=+2%2F%282%2A%285%2Fx%29%2B1%29 ALIGN=MIDDLE  ALT=\"g%28f%28x%29%29+=+2%2F%282%2A%285%2Fx%29%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in f(x) = 5/x. Simplify.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28f%28x%29%29+=+2%2F%2810%2Fx%2B1%29 ALIGN=MIDDLE  ALT=\"g%28f%28x%29%29+=+2%2F%2810%2Fx%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <font color=blue>(g o f)(x)</font> is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%29%2F%2810%2Bx%29 ALIGN=MIDDLE  ALT=\"%282x%29%2F%2810%2Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here we can't have x equal 0 or -10. Those values make the denominator zero. Any other value will work. \r<BR>\n" );
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document.write( "So the domain of (g o f)(x) is <font color=red>(-infinity, -10) U (-10, 0) U (0, infinity)</font> </SPAN>\n" );
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