document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=105421>Question 105421</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello everyone,\r<BR>\n" );
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document.write( "The question: Find the inverse of the function f(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt+%28x%5E2+%2B+%281%2Fx%5E2%29%29+ ALIGN=MIDDLE  ALT=\"+sqrt+%28x%5E2+%2B+%281%2Fx%5E2%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I am having trouble isolating the y for the inverse function. I would greatly appreciate any help. My work is shown below.\r<BR>\n" );
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document.write( "Thank you very much!\r<BR>\n" );
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document.write( "My work :\r<BR>\n" );
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document.write( "f(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt+%28x%5E2+%2B+%281%2Fx%5E2%29%29+ ALIGN=MIDDLE  ALT=\"+sqrt+%28x%5E2+%2B+%281%2Fx%5E2%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "y = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt+%28x%5E2+%2B+%281%2Fx%5E2%29%29+ ALIGN=MIDDLE  ALT=\"+sqrt+%28x%5E2+%2B+%281%2Fx%5E2%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt+%28y%5E2+%2B+%281%2Fy%5E2%29%29+ ALIGN=MIDDLE  ALT=\"+sqrt+%28y%5E2+%2B+%281%2Fy%5E2%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=+sqrt+%28%28y%5E4+%2B+1%29%2Fy%29+ ALIGN=MIDDLE  ALT=\"+sqrt+%28%28y%5E4+%2B+1%29%2Fy%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=+x%5E2+=+%28y%5E4+%2B+1%29%2Fy%5E2+ ALIGN=MIDDLE  ALT=\"+x%5E2+=+%28y%5E4+%2B+1%29%2Fy%5E2+\"   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=+x%5E2y%5E2+=+y%5E4+%2B+1+ ALIGN=MIDDLE  ALT=\"+x%5E2y%5E2+=+y%5E4+%2B+1+\"   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=+x%5E2y%5E2+-+y%5E4+=+1+ ALIGN=MIDDLE  ALT=\"+x%5E2y%5E2+-+y%5E4+=+1+\"   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=+y%5E2%28x%5E2+-+y%5E2%29+=+1+ ALIGN=MIDDLE  ALT=\"+y%5E2%28x%5E2+-+y%5E2%29+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #77038 by </B><A HREF=/tutors/aboutme.mpl?userid=TP><B>TP(29)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=TP><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=77038\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Fine up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt+%28y%5E2+%2B+1%2Fy%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt+%28y%5E2+%2B+1%2Fy%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So we'll go from there.<BR>\n" );
document.write( "x=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt+%28y%5E2+%2B+1%2Fy%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt+%28y%5E2+%2B+1%2Fy%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Square both sides:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=+y%5E2+%2B+1%2Fy%5E2+=+y%5E4%2Fy%5E2+%2B+1%2Fy%5E2+=+%28y%5E4+%2B+1%29%2Fy%5E2 ALIGN=MIDDLE  ALT=\"x%5E2=+y%5E2+%2B+1%2Fy%5E2+=+y%5E4%2Fy%5E2+%2B+1%2Fy%5E2+=+%28y%5E4+%2B+1%29%2Fy%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Now multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2x%5E2+=+y%5E4+%2B+1 ALIGN=MIDDLE  ALT=\"y%5E2x%5E2+=+y%5E4+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2x%5E2 ALIGN=MIDDLE  ALT=\"y%5E2x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides:<BR>\n" );
document.write( "We get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E4-x%5E2y%5E2+%2B+1+=+0 ALIGN=MIDDLE  ALT=\"y%5E4-x%5E2y%5E2+%2B+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (i)<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+y%5E2 ALIGN=MIDDLE  ALT=\"z+=+y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and (i) becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E2+-+x%5E2z+%2B+1+=+0 ALIGN=MIDDLE  ALT=\"z%5E2+-+x%5E2z+%2B+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now using the formula for the solution of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=az%5E2+%2B+bz+%2B+c+%2B+0 ALIGN=MIDDLE  ALT=\"az%5E2+%2B+bz+%2B+c+%2B+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > i.e.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-b%2B-sqrt%28b%5E2+-+4ac%29%29%2F2a ALIGN=MIDDLE  ALT=\"z+=+%28-b%2B-sqrt%28b%5E2+-+4ac%29%29%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b=-x%5E2 ALIGN=MIDDLE  ALT=\"+b=-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c=1 ALIGN=MIDDLE  ALT=\"+c=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28x%5E2%2B-sqrt%28x%5E4+-+4%29%29%2F2 ALIGN=MIDDLE  ALT=\"z+=+%28x%5E2%2B-sqrt%28x%5E4+-+4%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+y%5E2 ALIGN=MIDDLE  ALT=\"z+=+y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and so:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+=+%28x%5E2%2B-sqrt%28x%5E4+-+4%29%29%2F2 ALIGN=MIDDLE  ALT=\"y%5E2+=+%28x%5E2%2B-sqrt%28x%5E4+-+4%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Hence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+sqrt%28%28x%5E2%2B-sqrt%28x%5E4+-+4%29%29%2F2%29 ALIGN=MIDDLE  ALT=\"y=+sqrt%28%28x%5E2%2B-sqrt%28x%5E4+-+4%29%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Now there are TWO possible answers for the solution to the equation but only ONE is correct for the inverse function and it is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%28%28x%5E2+%2B+sqrt%28x%5E4+-+4%29%29%2F2%29 ALIGN=MIDDLE  ALT=\"y+=+sqrt%28%28x%5E2+%2B+sqrt%28x%5E4+-+4%29%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >ANS.<BR>\n" );
document.write( "(NB. I'll leave it up to you to work out what <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%28%28x%5E2+-+sqrt%28x%5E4+-+4%29%29%2F2%29 ALIGN=MIDDLE  ALT=\"y+=+sqrt%28%28x%5E2+-+sqrt%28x%5E4+-+4%29%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents.<BR>\n" );
document.write( "You should also use your calculator to check the inverse function answer - I suggest, for example, that you replace x by 4 or a similar small value).<BR>\n" );
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