document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1090122>Question 1090122</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 22. (See Week 5 Lecture page “Composite Functions” for a starting point.)<BR>\n" );
document.write( "The population of viruses in an influenza culture after t hours is given by the function<BR>\n" );
document.write( "x(t) = 4e^t/2.5<BR>\n" );
document.write( "The cost y in dollars for a new automated microscope to count x viruses in a sample is<BR>\n" );
document.write( "y(x) = ln 2x<BR>\n" );
document.write( "Create the composite function that calculates the cost y in dollars of counting the number of viruses in an influential culture after t hours. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #704516 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=704516\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> composite function is a function that operates on another function as its argument.\r<BR>\n" );
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document.write( "example:\r<BR>\n" );
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document.write( "f(x) = x^2\r<BR>\n" );
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document.write( "g(x) = 3x\r<BR>\n" );
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document.write( "g(f(x)) = 3 * x^2\r<BR>\n" );
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document.write( "the function f(x) replaces the argument of x in g(x).\r<BR>\n" );
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document.write( "you have x(t) = 4e^(t/2.5)\r<BR>\n" );
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document.write( "you have y(x) = ln(2x)\r<BR>\n" );
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document.write( "the function y(x(t)) replaces the argument of x in y(x) with the output of 4e^(t/2.5) in g(t)\r<BR>\n" );
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document.write( "this goes to the heart of the definition of a function.\r<BR>\n" );
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document.write( "in the function x(t) = 4e^(t/2.5):\r<BR>\n" );
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document.write( "x(t) is the name of the function.<BR>\n" );
document.write( "t is the input of the function.<BR>\n" );
document.write( "4e^(t/2.5) is the output of the function.\r<BR>\n" );
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document.write( "here's what i found on the web that looks like a pretty good reference.<BR>\n" );
document.write( "read it all the way through because it describes how to replace the original argument with another expression when evaluating that expression within the function.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://mathbitsnotebook.com/Algebra1/Functions/FNNotationEvaluation.html\" target = \"_blank\">https://mathbitsnotebook.com/Algebra1/Functions/FNNotationEvaluation.html</a>\r<BR>\n" );
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document.write( "here's also a reference on composite functions i found on the web.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.mesacc.edu/~scotz47781/mat120/notes/composition/composite_functions_intro.pdf\" target = \"_blank\">http://www.mesacc.edu/~scotz47781/mat120/notes/composition/composite_functions_intro.pdf</a>\r<BR>\n" );
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document.write( "i followed the rules in solving your problem.\r<BR>\n" );
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document.write( "the argument in the first function was replaced with the output of the second function.\r<BR>\n" );
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document.write( "once again, for ease of reference.\r<BR>\n" );
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document.write( "the first function is x(t) = 4e^(t/2.5)\r<BR>\n" );
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document.write( "the argument of this function is t and the output of this function is 4e^(t/2.5).\r<BR>\n" );
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document.write( "this tells you the population of viruses after t hours.\r<BR>\n" );
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document.write( "the second function is y(x) = ln(2x)\r<BR>\n" );
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document.write( "the argument of this function is x and the output of this function is ln(2x).\r<BR>\n" );
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document.write( "don't confuse the function name of x(t) with the argument of x in the function name of y(x).\r<BR>\n" );
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document.write( "they're two different things.\r<BR>\n" );
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document.write( "you could have named the function x(t) anything else and the function would have been the same.\r<BR>\n" );
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document.write( "some examples:\r<BR>\n" );
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document.write( "f(t) = 4e^(t/2.5)<BR>\n" );
document.write( "g(t) = 4e^(t/2.5)<BR>\n" );
document.write( "h(t) = 4e^(t/2.5)<BR>\n" );
document.write( "x(t) = 4e^(t/2.5)\r<BR>\n" );
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document.write( "they all point to the same relationship between the argument of t and the output of e^(t/2.5)\r<BR>\n" );
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document.write( "your other function of y(x) = ln(2x) could also have many names.\r<BR>\n" );
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document.write( "n(x) = ln(2x)<BR>\n" );
document.write( "p(x) = ln(2x)<BR>\n" );
document.write( "q(x) = ln(2x)<BR>\n" );
document.write( "y(x) = ln(2x)\r<BR>\n" );
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document.write( "all of these functions have an argument of x and an output of ln(2x).\r<BR>\n" );
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document.write( "back to your original problem.\r<BR>\n" );
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document.write( "your functions are:\r<BR>\n" );
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document.write( "x(t) = 4e^(t/2.5)\r<BR>\n" );
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document.write( "y(x) = ln(2x)\r<BR>\n" );
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document.write( "note the parentheses.<BR>\n" );
document.write( "they make it much clearer as to where each expression within the overall expression belong.<BR>\n" );
document.write( "hopefully i got it right.\r<BR>\n" );
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document.write( "x(t) represents the number of viruses after t hours<BR>\n" );
document.write( "y(x) represents the cost of a new microphone to count the number of viruses.\r<BR>\n" );
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document.write( "therefore y(x(t)) represents the cost of a new microphone to count the number of viruses after t hours.\r<BR>\n" );
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document.write( "the argument of y(x) is replaced by the output of the function x(t).\r<BR>\n" );
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document.write( "x(t) = 4e^(t/2.5)\r<BR>\n" );
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document.write( "y(x) = ln(2x)\r<BR>\n" );
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document.write( "y(x(t)) = ln(2 * 4e^(t/2.5))\r<BR>\n" );
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document.write( "i believe you can simplify that to y(x(t)) = ln(8e^(t/2.5))\r<BR>\n" );
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document.write( "that's what i get.\r<BR>\n" );
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document.write( "let me know if you have any further questions regarding this.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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