document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1092863>Question 1092863</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A.  Choose a composite trigonometric function of the form h(x) = g(f(x)) where the function g is one of the trigonometric functions and f is any function more complicated than y = x.<BR>\n" );
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document.write( "Note: An example of an appropriate function h would be h(x) = sin(3x – 4). However, you must pick a function other than this example for this task. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #707479 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=707479\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i picked h(x) = tan(x^2 + 2x + 1)\r<BR>\n" );
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document.write( "here's how that equation is derived.\r<BR>\n" );
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document.write( "you start with f(x) = x^2 + 2x + 1 and g(x) = tan(x).\r<BR>\n" );
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document.write( "you create h(x) = g(f(x)).\r<BR>\n" );
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document.write( "in g(x), the argument is x.\r<BR>\n" );
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document.write( "in g(f(x)), the argument of x is replaced with f(x) which means it is replaced with (x^2 + 2x + 1) because f(x) is equal to x^2 + 2x + 1.\r<BR>\n" );
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document.write( "therefore:\r<BR>\n" );
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document.write( "g(f(x)) becomes g(x^2 + 2x + 1) and, since g(x) is equal to tan(x), then g(f(x)) becomes tan(x^2 + 2x + 1).\r<BR>\n" );
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document.write( "to see how this works, we first start with f(x) = x^2 + 2x + 1.\r<BR>\n" );
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document.write( "a graph of this function shows the following:\r<BR>\n" );
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document.write( "when x = -5, y = 16<BR>\n" );
document.write( "when x = -1, y = 0<BR>\n" );
document.write( "when x = 5, y = 36\r<BR>\n" );
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document.write( "the graph is shown below:\r<BR>\n" );
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document.write( "we then work with g(x) = tan(x).\r<BR>\n" );
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document.write( "a graph of this function shows the following:\r<BR>\n" );
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document.write( "when x = -5, y = 3.381<BR>\n" );
document.write( "when x = -1, y = -1.557<BR>\n" );
document.write( "when x = 5, y = -3.381\r<BR>\n" );
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document.write( "the graph is shown below:\r<BR>\n" );
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document.write( "we then work with h(x) = g(f(x)) which results in h(x) = tan(x^2 + 2x + 1)\r<BR>\n" );
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document.write( "a graph of this function shows the following:\r<BR>\n" );
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document.write( "when x = -5, y = .301<BR>\n" );
document.write( "when x = -1, y = 0<BR>\n" );
document.write( "when x = 5, y = 7.75\r<BR>\n" );
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document.write( "the graph is shown below:\r<BR>\n" );
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document.write( "so what is happening?\r<BR>\n" );
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document.write( "we'll take one point to show the relationship.\r<BR>\n" );
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document.write( "when x = -5, f(x) = x^2 + 2x + 1 = (-5)^2 + 2*(-5) + 1 = 25 - 10 + 1 = 16\r<BR>\n" );
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document.write( "when x = -5, g(x) = tan(x) = tan(-5) = 3.381\r<BR>\n" );
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document.write( "when x = -5, h(x) = g(f(x)) = tan(x^2 + 2x + 1) = tan((-5)^2 + 2*(-5) + 1) = tan(25 - 10 + 1) = tan(16) = .301\r<BR>\n" );
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document.write( "you can check with your calculator.\r<BR>\n" );
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document.write( "when x = -5, calculate f(x) = x^2 + 2x + 1 and you will get 16.\r<BR>\n" );
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document.write( "when x = -5, calculate g(x) = tan(x) and you will get 3.380515006 which rounds to 3.381.\r<BR>\n" );
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document.write( "this particular solution has no relationship to f(x).<BR>\n" );
document.write( "it just shows how the tan(x) function looks on the graph.\r<BR>\n" );
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document.write( "when x = -5, calculate h(x) = g(f(x)) = tan(x^2 + 2x + 1) and you will get .300632242 which rounds to .301.\r<BR>\n" );
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document.write( "now find tan(16) and you will find that it is also equal to .300632242 which rounds to .301.\r<BR>\n" );
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document.write( "what happened is that, when x = -5, the function of g(f(x)) found the result of x = -5 in the function of f(x) and then calculated the tangent of that value.\r<BR>\n" );
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document.write( "so, when x = -5, h(x) found the tan of 16 which was the value of f(x) when x = -5.\r<BR>\n" );
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document.write( "you can follow the logic with the other points as well.\r<BR>\n" );
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document.write( "when x = 5, f(x) = 36)\r<BR>\n" );
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document.write( "when x = 5, h(x) = g(f(x)) = g(36) = tan(36) = 7.750470906which rounds to 7.75.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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