document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=856270>Question 856270</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the inverse of the one-to-one function.\r<BR>\n" );
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document.write( "f(x)=5x-3/2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #515846 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=515846\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> start with y = 5x - 3/2<BR>\n" );
document.write( "interchange y and x to get:<BR>\n" );
document.write( "x = 5y - 3/2<BR>\n" );
document.write( "solve for y as follows:<BR>\n" );
document.write( "add 3/2 to both sides of the equation to get:<BR>\n" );
document.write( "x + 3/2 = 5y<BR>\n" );
document.write( "commute this equation to get:<BR>\n" );
document.write( "5y = x + 3/2<BR>\n" );
document.write( "divide both sides of this equation by 5 to get:<BR>\n" );
document.write( "y = x/5 + 3/10\r<BR>\n" );
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document.write( "that should be your inverse function.<BR>\n" );
document.write( "if it is, then f(g(x)) should be equal to x and g(f(x)) should be equal to x.<BR>\n" );
document.write( "make your original equation f(x).<BR>\n" );
document.write( "make the inverse equation equal to g(x).<BR>\n" );
document.write( "you get:<BR>\n" );
document.write( "f(x)= 5x - 3/2<BR>\n" );
document.write( "g(x) = x/5 + 3/10\r<BR>\n" );
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document.write( "f(g(x)) is derived by replacing x in f(x) = 5x - 3/2 with (x/5 + 3/10) from g(x) to get:<BR>\n" );
document.write( "f(g(x)) = 5*(x/5 + 3/10) - 3/2<BR>\n" );
document.write( "simplify by distributing the multiplication to get:<BR>\n" );
document.write( "f(g(x)) = 5 * x/5 + 5 * 3/10 - 3/2<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "f(g(x)) = x + 15/10 - 3/2<BR>\n" );
document.write( "divide numerator and denominator of 15/10 by 5 to get:<BR>\n" );
document.write( "f(g(x)) = x + 3/2 - 3/2<BR>\n" );
document.write( "combine like terms to get:<BR>\n" );
document.write( "f(g(x)) = x<BR>\n" );
document.write( "one half of the test is done.<BR>\n" );
document.write( "now to the other half.\r<BR>\n" );
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document.write( "g(f(x)) is derived by replacing x in g(x) = x/5 + 3/10 with (5x - 3/2) from f(x) to get:<BR>\n" );
document.write( "g(f(x)) = (5x - 3/2) / 5 + 3/10<BR>\n" );
document.write( "multiply (5x - 3/2) / 5 * (2/2) to get:<BR>\n" );
document.write( "g(f(x)) = 2*(5x-3/2)/10 + 3/10<BR>\n" );
document.write( "combine the numerators under the common denominator to get:<BR>\n" );
document.write( "g(f(x)) = (2*(5x-3/2) + 3) / 10<BR>\n" );
document.write( "simplify by distributing the multiplication in the numerator to get:<BR>\n" );
document.write( "g(f(x)) = (10x - 3 + 3) / 10<BR>\n" );
document.write( "simplify by combining like terms to get:<BR>\n" );
document.write( "g(f(x)) = 10x / 10<BR>\n" );
document.write( "the 10 in the numerator and the 10 in the denominator cancel out and you are left with:<BR>\n" );
document.write( "g(f(x)) = x\r<BR>\n" );
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document.write( "the test for inverse functions is passed, so your solution is:\r<BR>\n" );
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document.write( "f(x) = 5x - 3/2<BR>\n" );
document.write( "g(x) = x/5 + 3/10 = inverse function of f(x)\r<BR>\n" );
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document.write( "the graphs of inverse functions are symmetric about the line y = x.<BR>\n" );
document.write( "the graph of these 2 functions is shown below:<BR>\n" );
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document.write( "they look symmetric.\r<BR>\n" );
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document.write( "another test is that f(x,y) is equal to g(y,x)\r<BR>\n" );
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document.write( "the terminology is scary and difficult to follow, but this is what happens:\r<BR>\n" );
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document.write( "start with:<BR>\n" );
document.write( "f(x) = 5x - 3/2 and replace f(x) with y to get:<BR>\n" );
document.write( "y = 5x - 3/2<BR>\n" );
document.write( "pick any value of x and solve for y.<BR>\n" );
document.write( "i picked x = 5 to get:<BR>\n" );
document.write( "y = 5(5) - 3/2 = 25 - 3/2 = 50/2 - 3/2 = 47/2.<BR>\n" );
document.write( "this results in:<BR>\n" );
document.write( "y = 47/2\r<BR>\n" );
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document.write( "now look at g(x) = x/5 + 3/10 and replace g(x) with y to get:<BR>\n" );
document.write( "y = x/5 + 3/10<BR>\n" );
document.write( "now replace x in this equation with y from the first equation to get:<BR>\n" );
document.write( "y = (47/2)/5 + 3/10<BR>\n" );
document.write( "simplify this to get:<BR>\n" );
document.write( "y = 47/10 + 3/10<BR>\n" );
document.write( "combine the numerators under the common denominator of 10 to get:<BR>\n" );
document.write( "y = 50/10 which is equal to 5.<BR>\n" );
document.write( "you get y = 5\r<BR>\n" );
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document.write( "f(x,y) is equal to (5,47/2)\r<BR>\n" );
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document.write( "g(x,y) is equal to (47/2,5)\r<BR>\n" );
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document.write( "the x from g(x) is equal to the y from f(x).<BR>\n" );
document.write( "the y from g(x) is equal to the x from f(x).\r<BR>\n" );
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document.write( "this is what they mean when they say f(x,y) = g(y,x).\r<BR>\n" );
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document.write( "you take the output from f(x) and make it the input to g(x) and the result is you are back to x.\r<BR>\n" );
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document.write( "5 becomes 47/2 after being put through f(x) and then 47/2 becomes 5 after being put through g(x).\r<BR>\n" );
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document.write( "the inverse function undoes what the original function does.\r<BR>\n" );
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document.write( "f(x) makes 5 become 47/2 and g(x) undoes that by making 47/2 become 5.<BR>\n" );
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