document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196464>Question 1196464</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find a formula for the inverse of the function.<BR>\n" );
document.write( "f(x) = 6x − 1/2x + 9\r<BR>\n" );
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document.write( "f −1(x) =       (f inverse of x).\r<BR>\n" );
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document.write( "  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #829316 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13198)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=829316\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "If you are working on a problem like this, your knowledge of math should be enough for you to know that the proper use of parentheses is important.  The function as you show it is this:<br><BR>\n" );
document.write( "f(x) = 6x − 1/2x + 9  -->  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+6x+-+1%2F2x+%2B+9 ALIGN=MIDDLE  ALT=\"f%28x%29+=+6x+-+1%2F2x+%2B+9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That is almost certainly not what you intended; you probably meant<br><BR>\n" );
document.write( "f(x) = (6x - 1)/(2x + 9)  -->  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%286x+-+1%29%2F%282x+%2B+9%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%286x+-+1%29%2F%282x+%2B+9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Finding the inverse of a function like this is a common type of problem on high school math competition tests; there is an easy pattern that makes it easy to write down the inverse function without doing any work.<br><BR>\n" );
document.write( "Let's take a generic example <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28Ax%2BB%29%2F%28Cx%2BD%29 ALIGN=MIDDLE  ALT=\"y=%28Ax%2BB%29%2F%28Cx%2BD%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and find the inverse by the standard method of switching x and y and solving for the new y.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28Ay%2BB%29%2F%28Cy%2BD%29 ALIGN=MIDDLE  ALT=\"x=%28Ay%2BB%29%2F%28Cy%2BD%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28Cy%2BD%29=Ay%2BB ALIGN=MIDDLE  ALT=\"x%28Cy%2BD%29=Ay%2BB\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Cxy%2BDx=Ay%2BB ALIGN=MIDDLE  ALT=\"Cxy%2BDx=Ay%2BB\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Cxy-Ay=-Dx%2BB ALIGN=MIDDLE  ALT=\"Cxy-Ay=-Dx%2BB\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%28Cx-A%29=-Dx%2BB ALIGN=MIDDLE  ALT=\"y%28Cx-A%29=-Dx%2BB\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-Dx%2BB%29%2F%28Cx-A%29 ALIGN=MIDDLE  ALT=\"y=%28-Dx%2BB%29%2F%28Cx-A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now compare the original function to its inverse function:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28Ax%2BB%29%2F%28Cx%2BD%29 ALIGN=MIDDLE  ALT=\"y=%28Ax%2BB%29%2F%28Cx%2BD%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-Dx%2BB%29%2F%28Cx-A%29 ALIGN=MIDDLE  ALT=\"y=%28-Dx%2BB%29%2F%28Cx-A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In the inverse, the B and C stay where they are, while the A and D switch places and both change sign.<br><BR>\n" );
document.write( "Using this pattern, you can write down the inverse of your given function with little effort:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%286x-1%29%2F%282x%2B9%29 ALIGN=MIDDLE  ALT=\"y=%286x-1%29%2F%282x%2B9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-9x-1%29%2F%282x-6%29 ALIGN=MIDDLE  ALT=\"%28-9x-1%29%2F%282x-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
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