document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1054920>Question 1054920</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need the inverse of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%283x-2%29%2F%285x-3%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%283x-2%29%2F%285x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with proof (like, f(f^-1(x)) that it's the inverse. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #670125 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=670125\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let y = f(x).\r<BR>\n" );
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document.write( "your equation becomes y = (3x-2) / (5x-3)\r<BR>\n" );
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document.write( "replace y with x and x with y to get x = (3y-2) / (5y-3)\r<BR>\n" );
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document.write( "multiply both sides of the equation by (5y-3) to get x * (5y-3) = (3y-2)\r<BR>\n" );
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document.write( "simplify to get 5xy - 3x = 3y - 2\r<BR>\n" );
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document.write( "subtract 3y from both sides of the equation and add 3x to both sides of the equation to get 5xy - 3y = 3x - 2\r<BR>\n" );
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document.write( "factor out the y on the left hand side of the equation to get y * (5x - 3) =  3x - 2\r<BR>\n" );
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document.write( "divide both sides of the equaton by (5x - 3) to get y = (3x - 2) / (5x - 3).\r<BR>\n" );
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document.write( "your original equation was y = (3x - 2) / (5x - 3).\r<BR>\n" );
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document.write( "your inverse equation is y = (3x - 2) / (5x -3).\r<BR>\n" );
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document.write( "they're the same equation !!!!!.\r<BR>\n" );
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document.write( "is that possible?\r<BR>\n" );
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document.write( "if they are inverse equations of each other, then they will be reflections about the line y = x.\r<BR>\n" );
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document.write( "the following graph confirms that this relationship is true.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/102905.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "if they are inverse equations, then f(x,y) = g(y,x).\r<BR>\n" );
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document.write( "when x = 0, the value of y in the original equation becomes (3*0-2) / (5*0-3) which is equal to -2/-3 which is equal to 2/3.\r<BR>\n" );
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document.write( "when x = 2/3, the value of y in the inverse equation becomes (3*2/3 - 2) / (5*2/3 - 3) which becomes (2-2) / (10/3 - 3) which becomes 0 / (1/3) which is equal to 0.\r<BR>\n" );
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document.write( "therefore f(x,y) = 2/3 when x = 0 and g(y,x) becomes 0 when y = 2/3.<BR>\n" );
document.write( "this confirms that the same equation is the inverse equation of itself.\r<BR>\n" );
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document.write( "the last test is fog(x) = c and gof(x) = x\r<BR>\n" );
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document.write( "f(x) = (3x-2)/(5x-3)<BR>\n" );
document.write( "g(x) = (3x-2)/(5x-3)<BR>\n" );
document.write( "fog(x) = f(g(x)) = f((3x-2)/(5x-3))\r<BR>\n" );
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document.write( "f((3x-2)/(5x-3)) is equal to (3*(3x-2)/(5x-3) -2) / (5*(3x-2)/(5x-3) - 3))\r<BR>\n" );
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document.write( "the following worksheet shows the details of the calculations.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/102906.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "you wind up with f(g(x)) = x which passes the test for inverse functions.\r<BR>\n" );
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document.write( "if you did g(f(x)) you would have wound up with the same equation, so g(f(x)) is automatically equal to x also.\r<BR>\n" );
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document.write( "that's the final proof that f(x) and g(x) are inverse equations.\r<BR>\n" );
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document.write( "they are also identical equations.\r<BR>\n" );
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