document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170515>Question 1170515</A>: <I><SPAN STYLE=\"word-break: break-all;\"> f  and g are one to one functions and satisfy<BR>\n" );
document.write( "f(3)=7,f(11)=16,f(7)=11,f(16)=3,g(7)=16,g(11)=7,g(3)=11,g(16)=3\r<BR>\n" );
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document.write( "Find<BR>\n" );
document.write( "((f∘g)(16))−1−((g∘f)(16))−1\r<BR>\n" );
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document.write( "Answer: <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851160 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851160\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step by step:\r<BR>\n" );
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document.write( "**1. Understand Composition of Functions**\r<BR>\n" );
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document.write( "* (f∘g)(x) = f(g(x))<BR>\n" );
document.write( "* (g∘f)(x) = g(f(x))\r<BR>\n" );
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document.write( "**2. Calculate (f∘g)(16)**\r<BR>\n" );
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document.write( "* (f∘g)(16) = f(g(16))<BR>\n" );
document.write( "* We're given g(16) = 3<BR>\n" );
document.write( "* So (f∘g)(16) = f(3)<BR>\n" );
document.write( "* We're given f(3) = 7<BR>\n" );
document.write( "* Therefore, (f∘g)(16) = 7\r<BR>\n" );
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document.write( "**3. Calculate (g∘f)(16)**\r<BR>\n" );
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document.write( "* (g∘f)(16) = g(f(16))<BR>\n" );
document.write( "* We're given f(16) = 3<BR>\n" );
document.write( "* So (g∘f)(16) = g(3)<BR>\n" );
document.write( "* We're given g(3) = 11<BR>\n" );
document.write( "* Therefore, (g∘f)(16) = 11\r<BR>\n" );
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document.write( "**4. Find ((f∘g)(16))−1**\r<BR>\n" );
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document.write( "* ((f∘g)(16))−1 = 7−1<BR>\n" );
document.write( "* We want to find the inverse of 7.\r<BR>\n" );
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document.write( "* To find f^-1(7), we look for a value x such that f(x)=7. f(3)=7. therefore f^-1(7) = 3.<BR>\n" );
document.write( "* To find g^-1(11), we look for a value x such that g(x)=11. g(3)=11. therefore g^-1(11)=3.\r<BR>\n" );
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document.write( "* ((f∘g)(16))−1 = f^-1(7) = 3\r<BR>\n" );
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document.write( "**5. Find ((g∘f)(16))−1**\r<BR>\n" );
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document.write( "* ((g∘f)(16))−1 = 11−1<BR>\n" );
document.write( "* To find f^-1(11), we look for a value x such that f(x)=11. f(7)=11. therefore f^-1(11) = 7.<BR>\n" );
document.write( "* To find g^-1(11), we look for a value x such that g(x)=11. g(3)=11. therefore g^-1(11)=3.<BR>\n" );
document.write( "* ((g∘f)(16))−1 = g^-1(11) = 3.\r<BR>\n" );
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document.write( "**6. Calculate the Final Result**\r<BR>\n" );
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document.write( "* ((f∘g)(16))−1 − ((g∘f)(16))−1 = 3 - 3 = 0\r<BR>\n" );
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document.write( "**Final Answer:** The final answer is 0.<BR>\n" );
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