document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206605>Question 1206605</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the equation defines a one-to-one function f.<BR>\n" );
document.write( "f(x) = 4x − 1<BR>\n" );
document.write( "Verify that <BR>\n" );
document.write( "f ∘ f −1<BR>\n" );
document.write( " and <BR>\n" );
document.write( "f −1 ∘ f<BR>\n" );
document.write( " are both the identity function.<BR>\n" );
document.write( "(f ∘ f −1)(x)	 = 	f(f −1(x))\r<BR>\n" );
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document.write( "(f −1 ∘ f)(x)	 = 	f −1(f(x)) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #844159 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=844159\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+4x+%E2%88%92+1 ALIGN=MIDDLE  ALT=\"f%28x%29+=+4x+%E2%88%92+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Verify that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" > ∘ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%E2%80%89%5E-1 ALIGN=MIDDLE  ALT=\"f%E2%80%89%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E-1 ALIGN=MIDDLE  ALT=\"f%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ∘ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "are both the identity function.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f+ ALIGN=MIDDLE  ALT=\"f+\"   DISABLED_style= \"max-width:90%; height:auto;\" >∘ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E-1 ALIGN=MIDDLE  ALT=\"f%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29+=+f%28f%5E-1%28x%29%29 ALIGN=MIDDLE  ALT=\"%28x%29+=+f%28f%5E-1%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E-1 ALIGN=MIDDLE  ALT=\"f%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ∘ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29+=+f%5E-1%28f%28x%29%29 ALIGN=MIDDLE  ALT=\"%28x%29+=+f%5E-1%28f%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E-1 ALIGN=MIDDLE  ALT=\"f%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+4x+-+1 ALIGN=MIDDLE  ALT=\"f%28x%29+=+4x+-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >...<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+y ALIGN=MIDDLE  ALT=\"f%28x%29+=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+4x+-1 ALIGN=MIDDLE  ALT=\"y=+4x+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >...swap variables\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=+4y+-1 ALIGN=MIDDLE  ALT=\"x=+4y+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >...solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1=+4y ALIGN=MIDDLE  ALT=\"x%2B1=+4y\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%2F4%2B1%2F4 ALIGN=MIDDLE  ALT=\"y=x%2F4%2B1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "=>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E-1%28x%29=x%2F4%2B1%2F4 ALIGN=MIDDLE  ALT=\"f%5E-1%28x%29=x%2F4%2B1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f+ ALIGN=MIDDLE  ALT=\"f+\"   DISABLED_style= \"max-width:90%; height:auto;\" >∘ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E-1 ALIGN=MIDDLE  ALT=\"f%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29+=f%28f%5E-1%29=f%28x%2F4%2B1%2F4%29=4%28x%2F4%2B1%2F4%29+-+1=x%2B1+-+1=x ALIGN=MIDDLE  ALT=\"%28x%29+=f%28f%5E-1%29=f%28x%2F4%2B1%2F4%29=4%28x%2F4%2B1%2F4%29+-+1=x%2B1+-+1=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E-1 ALIGN=MIDDLE  ALT=\"f%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ∘ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29+=f%5E-1%284x-1%29=%284x-1%29%2F4%2B1%2F4=4x%2F4-1%2F4%2B1%2F4=x ALIGN=MIDDLE  ALT=\"%28x%29+=f%5E-1%284x-1%29=%284x-1%29%2F4%2B1%2F4=4x%2F4-1%2F4%2B1%2F4=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "proven that<BR>\n" );
document.write( "(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f+ ALIGN=MIDDLE  ALT=\"f+\"   DISABLED_style= \"max-width:90%; height:auto;\" >∘ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E-1 ALIGN=MIDDLE  ALT=\"f%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29+=+f%28f%5E-1%28x%29%29=x ALIGN=MIDDLE  ALT=\"%28x%29+=+f%28f%5E-1%28x%29%29=x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E-1 ALIGN=MIDDLE  ALT=\"f%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ∘ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29+=+f%5E-1%28f%28x%29%29=x ALIGN=MIDDLE  ALT=\"%28x%29+=+f%5E-1%28f%28x%29%29=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );