document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=481128>Question 481128</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The one-to-one function is defined by\r<BR>\n" );
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document.write( "f(x) = x/(8-9x)\r<BR>\n" );
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document.write( "Find  f^-1, the inverse of f . Then, give the domain and range of f^-1 using interval notation. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #329474 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=329474\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let y = f(x) to get:<BR>\n" );
document.write( "y = x/(8-9x)<BR>\n" );
document.write( "make y = x and x = y to get:<BR>\n" );
document.write( "x = y/(8-9y)<BR>\n" );
document.write( "multiply both sides of this equation by (8-9y) to get:<BR>\n" );
document.write( "x(8-9y) = y<BR>\n" );
document.write( "perform the indicated operation to get:<BR>\n" );
document.write( "8x - 9xy = y<BR>\n" );
document.write( "add 9xy to both sides of this equation to get:<BR>\n" );
document.write( "8x = 9xy + y<BR>\n" );
document.write( "factor out the y to get:<BR>\n" );
document.write( "8x = y(9x+1)<BR>\n" );
document.write( "divide both sides of this equation by (9x+1) to get:<BR>\n" );
document.write( "8x/(9x+1) = y<BR>\n" );
document.write( "commute this equation to get:<BR>\n" );
document.write( "y = 8x/(9x+1)<BR>\n" );
document.write( "your original equation is:<BR>\n" );
document.write( "y = x/(8-9x)<BR>\n" );
document.write( "your inverse equation is:<BR>\n" );
document.write( "y = 8x/(9x+1)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "if y = 8x/(1+9x) is an inverse equation of y = x/(8-9x), then the coordinate pairs of your original equation will be reversed from the coordinate pairs of your inverse equation.<BR>\n" );
document.write( "this means that:<BR>\n" );
document.write( "(x,y) = (a,b) in the original equation will become (x,y) = (b,a) in the inverse equation.<BR>\n" );
document.write( "an example:<BR>\n" );
document.write( "let x = 5 in your original equation.<BR>\n" );
document.write( "solve the original equation to get y = -5/37.<BR>\n" );
document.write( "your coordinate pairs from the original equation are (x,y) = (5,-5/37).<BR>\n" );
document.write( "now let x = -5/37 in your inverse equation.<BR>\n" );
document.write( "solve the inverse equation to get y = 5.<BR>\n" );
document.write( "your coordinate pairs from the inverse equation are (x,y) = (-5/37,5).<BR>\n" );
document.write( "since x,y = (a,b) in your original equation and x,y = (b,a) in you inverse equation, then this is a confirmation that your inverse equation has been calculated correctly.<BR>\n" );
document.write( "graph both equations.<BR>\n" );
document.write( "if you graph your original equation and your inverse equation, they should look like a reflection about the line y = x.<BR>\n" );
document.write( "the following graph shows both equations and also shows the equation of y = x.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-5%2C5%2C-5%2C5%2Cx%2F%288-9x%29%2C8x%2F%281%2B9x%29%2Cx%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-5%2C5%2C-5%2C5%2Cx%2F%288-9x%29%2C8x%2F%281%2B9x%29%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "you can see that the graph of the original equation and the graph of the inverse equation look like reflections about the line y = x.<BR>\n" );
document.write( "note that the original equation appears to have an asymptote at somewhere around x = 1 and that the inverse equation appears to have an asymptote at somewhere around y = 1.<BR>\n" );
document.write( "the asymptote for the original equation is really at x = 8/9.<BR>\n" );
document.write( "when that happens, the original equation of y = x/(8-9x) gets a denominator of 0 which means that the value of y is undefined which means there is a vertical asymptote at x = 8/9.<BR>\n" );
document.write( "the vertical asymptote for the original equation should become the horizontal asymptote for the inverse equation.<BR>\n" );
document.write( "this means that the inverse equation should have a horizontal asymptote at y = (8/9).<BR>\n" );
document.write( "the easiest way to find the horizontal asymptote is to solve the inverse equation for x.<BR>\n" );
document.write( "when you do that, you get back to the original equation, with the exception that x = y and y = x.<BR>\n" );
document.write( "you wind up with x = y/(8-9y) which leads to an asymptote of y = 8/9.<BR>\n" );
document.write( "the domain of your original equation is equal to all real values of x except x = 8/9.<BR>\n" );
document.write( "the range of your original equation is all real values of y.<BR>\n" );
document.write( "the domain of your inverse equation is equal to all real values of x.<BR>\n" );
document.write( "the range of your inverse equation is equal to all real values of y except y = 8/9.<BR>\n" );
document.write( "in the next graph, a vertical line was drawn at x = 8/9 (as best i can produce it using the algebra.com graphing software) and a horizontal line was drawn at y = 8/9, to show you where the vertical and horizontal asymptotes are.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-5%2C5%2C-5%2C5%2Cx%2F%288-9x%29%2C8x%2F%281%2B9x%29%2Cx%2C%288%2F9%29%2C100%28x-%288%2F9%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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