document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1125606>Question 1125606</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Analytically show that the function f(x)=10-&#8731;(x-8)<BR>\n" );
document.write( "is one-to-one, find its inverse, and evaluate the following:\r<BR>\n" );
document.write( "\n" );
document.write( "	f^(-1) (10)\r<BR>\n" );
document.write( "\n" );
document.write( "	f^(-1) (11)\r<BR>\n" );
document.write( "\n" );
document.write( "	f^(-1) (12)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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 #741902 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=741902\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\"><BR>\n" );
document.write( "If f(x) is one-to-one, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28a%29=f%28b%29 ALIGN=MIDDLE  ALT=\"f%28a%29=f%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > implies that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b ALIGN=MIDDLE  ALT=\"a+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all a,b in the domain of f(x). Put in plain english: if two output values are the same, then the inputs must be the same for the function to be one-to-one.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Consider a counter-example such as a parabola. We can have the same output lead to two different inputs (eg: y = 4 lead to x = -2 and x = 2 for the function y = x^2). This is a reason why a parabola is not one-to-one.\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%28x%29+=+10-root%283%2Cx-8%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+10-root%283%2Cx-8%29\"   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%28a%29+=+10-root%283%2Ca-8%29 ALIGN=MIDDLE  ALT=\"f%28a%29+=+10-root%283%2Ca-8%29\"   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%28b%29+=+10-root%283%2Cb-8%29 ALIGN=MIDDLE  ALT=\"f%28b%29+=+10-root%283%2Cb-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28a%29+=+f%28b%29 ALIGN=MIDDLE  ALT=\"f%28a%29+=+f%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then...\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%28a%29+=+f%28b%29 ALIGN=MIDDLE  ALT=\"f%28a%29+=+f%28b%29\"   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=10-root%283%2Ca-8%29+=+10-root%283%2Cb-8%29 ALIGN=MIDDLE  ALT=\"10-root%283%2Ca-8%29+=+10-root%283%2Cb-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Substitution\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-root%283%2Ca-8%29+=+-root%283%2Cb-8%29 ALIGN=MIDDLE  ALT=\"-root%283%2Ca-8%29+=+-root%283%2Cb-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 10 from both sides\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2Ca-8%29+=+root%283%2Cb-8%29 ALIGN=MIDDLE  ALT=\"root%283%2Ca-8%29+=+root%283%2Cb-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by -1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-8+=+b-8 ALIGN=MIDDLE  ALT=\"a-8+=+b-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Cube both sides\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b ALIGN=MIDDLE  ALT=\"a+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 8 to each side\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So we end up with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b ALIGN=MIDDLE  ALT=\"a+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > after assuming <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28a%29+=+f%28b%29 ALIGN=MIDDLE  ALT=\"f%28a%29+=+f%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28a%29+=+f%28b%29 ALIGN=MIDDLE  ALT=\"f%28a%29+=+f%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > leads to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b ALIGN=MIDDLE  ALT=\"a+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If we follow the steps shown above in reverse, then we'll go from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b ALIGN=MIDDLE  ALT=\"a+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28a%29+=+f%28b%29 ALIGN=MIDDLE  ALT=\"f%28a%29+=+f%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore, we have proven that f(x) is indeed one-to-one.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's find the inverse. Which I'll call g(x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "g(x) = f^(-1)(x)\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%28x%29+=+10-root%283%2Cx-8%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+10-root%283%2Cx-8%29\"   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=y+=+10-root%283%2Cx-8%29 ALIGN=MIDDLE  ALT=\"y+=+10-root%283%2Cx-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace f(x) with y\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+10-root%283%2Cy-8%29 ALIGN=MIDDLE  ALT=\"x+=+10-root%283%2Cy-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Swap x and y. From here on out, we're solving for y.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-10+=+-root%283%2Cy-8%29 ALIGN=MIDDLE  ALT=\"x-10+=+-root%283%2Cy-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 10 from both sides\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-root%283%2Cy-8%29+=+x-10 ALIGN=MIDDLE  ALT=\"-root%283%2Cy-8%29+=+x-10\"   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=root%283%2Cy-8%29+=+-x%2B10 ALIGN=MIDDLE  ALT=\"root%283%2Cy-8%29+=+-x%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by -1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8+=+%28-x%2B10%29%5E3 ALIGN=MIDDLE  ALT=\"y-8+=+%28-x%2B10%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Cube both sides\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-x%2B10%29%5E3%2B8 ALIGN=MIDDLE  ALT=\"y+=+%28-x%2B10%29%5E3%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 8 to both sides\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The inverse is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+%28-x%2B10%29%5E3%2B8 ALIGN=MIDDLE  ALT=\"g%28x%29+=+%28-x%2B10%29%5E3%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's use the inverse to compute the inputs x = 10, x = 11, x = 12\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+%28-x%2B10%29%5E3%2B8 ALIGN=MIDDLE  ALT=\"g%28x%29+=+%28-x%2B10%29%5E3%2B8\"   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=g%2810%29+=+%28-10%2B10%29%5E3%2B8 ALIGN=MIDDLE  ALT=\"g%2810%29+=+%28-10%2B10%29%5E3%2B8\"   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=g%2810%29+=+%280%29%5E3%2B8 ALIGN=MIDDLE  ALT=\"g%2810%29+=+%280%29%5E3%2B8\"   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=g%2810%29+=+0%2B8 ALIGN=MIDDLE  ALT=\"g%2810%29+=+0%2B8\"   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=g%2810%29+=+8 ALIGN=MIDDLE  ALT=\"g%2810%29+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+%28-x%2B10%29%5E3%2B8 ALIGN=MIDDLE  ALT=\"g%28x%29+=+%28-x%2B10%29%5E3%2B8\"   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=g%2811%29+=+%28-11%2B10%29%5E3%2B8 ALIGN=MIDDLE  ALT=\"g%2811%29+=+%28-11%2B10%29%5E3%2B8\"   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=g%2811%29+=+%28-1%29%5E3%2B8 ALIGN=MIDDLE  ALT=\"g%2811%29+=+%28-1%29%5E3%2B8\"   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=g%2811%29+=+-1%2B8 ALIGN=MIDDLE  ALT=\"g%2811%29+=+-1%2B8\"   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=g%2811%29+=+7 ALIGN=MIDDLE  ALT=\"g%2811%29+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+%28x%2B10%29%5E3%2B8 ALIGN=MIDDLE  ALT=\"g%28x%29+=+%28x%2B10%29%5E3%2B8\"   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=g%2812%29+=+%28-12%2B10%29%5E3%2B8 ALIGN=MIDDLE  ALT=\"g%2812%29+=+%28-12%2B10%29%5E3%2B8\"   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=g%2812%29+=+%28-2%29%5E3%2B8 ALIGN=MIDDLE  ALT=\"g%2812%29+=+%28-2%29%5E3%2B8\"   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=g%2812%29+=+-8%2B8 ALIGN=MIDDLE  ALT=\"g%2812%29+=+-8%2B8\"   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=g%2812%29+=+0 ALIGN=MIDDLE  ALT=\"g%2812%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );