document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209327>Question 1209327</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let<BR>\n" );
document.write( "f(x) = (2x + 5)/(x - 4).<BR>\n" );
document.write( "If f^{-1} is the inverse of f, what is f^{-1}(1)? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848377 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=848377\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Sometimes inverse notation can be a pain to work with. Especially with a keyboard.<BR>\n" );
document.write( "I find it's better to introduce another function such as g(x)\r<BR>\n" );
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document.write( "Let g(x) be the inverse of f(x)<BR>\n" );
document.write( "To find g(x), we first replace f(x) with y.<BR>\n" );
document.write( "Then swap x and y and solve for y.\r<BR>\n" );
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document.write( "f(x) = (2x + 5)/(x - 4)<BR>\n" );
document.write( "y = (2x + 5)/(x - 4)<BR>\n" );
document.write( "x = (2y + 5)/(y - 4) ...... x and y swap; from here we isolate y.<BR>\n" );
document.write( "x(y-4) = 2y + 5<BR>\n" );
document.write( "xy-4x = 2y+5<BR>\n" );
document.write( "xy-2y = 5+4x<BR>\n" );
document.write( "y(x-2) = 5+4x<BR>\n" );
document.write( "y = (5+4x)/(x-2)<BR>\n" );
document.write( "<font color=blue>g(x) = (5+4x)/(x-2) is the inverse of f(x)</font>\r<BR>\n" );
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document.write( "To confirm that f and g are inverses of each other, you should prove that<BR>\n" );
document.write( "f( g(x) ) = x and f( g(x) ) = x<BR>\n" );
document.write( "are both true equations for all x in the domain.<BR>\n" );
document.write( "I'll let the student handle this proof.\r<BR>\n" );
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document.write( "Once we figure out the inverse, we can then wrap up the question<BR>\n" );
document.write( "g(x) = (5+4x)/(x-2)<BR>\n" );
document.write( "g(1) = (5+4*1)/(1-2)<BR>\n" );
document.write( "g(1) = (9)/(-1)<BR>\n" );
document.write( "g(1) = <font color=red>-9 is the final answer.</font>\r<BR>\n" );
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document.write( "This is equivalent to saying f^{-1}(1) = <font color=red>-9</font> i.e. <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large f^{-1}(1) = \"><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large {\color{red}{-9}}\"><BR>\n" );
document.write( "But again the -1 exponent notation might be a bit clunky to write out on a keyboard.\r<BR>\n" );
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document.write( "Another approach\r<BR>\n" );
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document.write( "The input x maps to the output f(x) when applying the f(x) function.\r<BR>\n" );
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document.write( "The inverse goes in reverse of this process.<BR>\n" );
document.write( "Computing f^{-1}(1) is the same as asking \"what x value gives the output y = f(x) = 1?\"\r<BR>\n" );
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document.write( "Replace f(x) with 1 and solve for x.<BR>\n" );
document.write( "f(x) = (2x + 5)/(x - 4)<BR>\n" );
document.write( "1 = (2x + 5)/(x - 4)<BR>\n" );
document.write( "x-4 = 2x+5<BR>\n" );
document.write( "x-2x = 5+4<BR>\n" );
document.write( "-x = 9<BR>\n" );
document.write( "x = <font color=red>-9</font> is the x value input needed to arrive at f(x) = 1\r<BR>\n" );
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document.write( "Let's check that claim:<BR>\n" );
document.write( "f(x) = (2x + 5)/(x - 4)<BR>\n" );
document.write( "f(-9) = (2*(-9) + 5)/(-9 - 4)<BR>\n" );
document.write( "f(-9) = (-18 + 5)/(-13)<BR>\n" );
document.write( "f(-9) = (-13)/(-13)<BR>\n" );
document.write( "f(-9) = 1<BR>\n" );
document.write( "This verifies the answer.<BR>\n" );
document.write( "You can also use graphing tools like <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a> and <a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a> to verify.\r<BR>\n" );
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document.write( "Answer: <font color=red>-9</font><BR>\n" );
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