document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1124860>Question 1124860</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For each of the following functions, write the formula for the function's inverse. \r<BR>\n" );
document.write( "\n" );
document.write( "a. f(x)=4^x where y=f(x).\r<BR>\n" );
document.write( "\n" );
document.write( "f^-1(y)= ?\r<BR>\n" );
document.write( "\n" );
document.write( "b. f(x)=4(2.8)^x where y=f(x)\r<BR>\n" );
document.write( "\n" );
document.write( "f^-1(y)\r<BR>\n" );
document.write( "\n" );
document.write( "c. f(x)= log(x/19) where y=f(x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "f^-1(y)= ?<BR>\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 #741164 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=741164\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a. f(x)=4^x where y=f(x). <BR>\n" );
document.write( "f^-1(y)= ?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "let y = 4^x<BR>\n" );
document.write( "replace x with y and y with x to get x = 4^y.<BR>\n" );
document.write( "take the log of both sides of this equationt to get log(x) = log(4^y). <BR>\n" );
document.write( "since log(4^y) = y * log(4), your equation becomes log(x) = y * log(4).<BR>\n" );
document.write( "divide both sides of this equation by log(4) to get log(x) / log(4) = y<BR>\n" );
document.write( "that's your inverse equation.<BR>\n" );
document.write( "you get:<BR>\n" );
document.write( "y = 4^x<BR>\n" );
document.write( "y^-1 = log(x) / log(4)<BR>\n" );
document.write( "if y = f(x), then y^-1 = f^-1(x) which is what you called f^-1(y).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "b. f(x)=4(2.8)^x where y=f(x) <BR>\n" );
document.write( "f^-1(y) \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "let y = 4 * 2.8^x<BR>\n" );
document.write( "replace x with y and y with x to get x = 4 * 2.8^y<BR>\n" );
document.write( "take the log of both sides of this equation to get log(x) = log(4 * 2.8^y)<BR>\n" );
document.write( "since log(4 * 2.8^y) = log(4) + log(2.8^y) and since log(2.8^y) = y * log(2.8), this equation becomes log(x) = log(4) + y * log(2.8)<BR>\n" );
document.write( "subtract log(4) from both sides of this equation to get log(x) - log(4) = y * log(2.8)<BR>\n" );
document.write( "since log(x) - log(4) = log(x/4), this equation becomes log(x/4) = y * log(2.8).<BR>\n" );
document.write( "divide both sides of this equation by log(2.8) to get log(x/4) / log(2.8) = y.<BR>\n" );
document.write( "that's your inverse equation.<BR>\n" );
document.write( "you get y = 4 * 2.8^x<BR>\n" );
document.write( "y^-1 = log(x/4) / log(2.8)<BR>\n" );
document.write( "if y = f(x) the y^-1 = f^-1(x) which is what you call f^-1(y).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "c. f(x)= log(x/19) where y=f(x)<BR>\n" );
document.write( "f^-1(y)= ?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "let y = log(x/19)<BR>\n" );
document.write( "replace x with y and y with x to get x = log(y/19)<BR>\n" );
document.write( "log, by itself, means log10 which means log to the base of 10.<BR>\n" );
document.write( "this equation can therefore be shown as x = log10(y/19).<BR>\n" );
document.write( "the basic definition of logs says that x = logb(y) if and only if y = b^x.<BR>\n" );
document.write( "in your problem, this translates to x = log10(y/19) if and only if y/19 = 10^x.<BR>\n" );
document.write( "multiply both sides of this equation by 19 to get y = 19 * 10^x.<BR>\n" );
document.write( "that's your inverse equation.<BR>\n" );
document.write( "you get y = log(x/19)<BR>\n" );
document.write( "y^-1 = 19 * 10^x<BR>\n" );
document.write( "if y = f(x) the y^-1 = f^-1(x) which is what you call f^-1(y).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "one way to determine if these equations are really inverse equations is to graph them about the line y = x.<BR>\n" );
document.write( "if you then take a perpendicular to the line y = x, then the intersection of the regular equation and the inverse equation to this vertical line will be (x,y) on one side and (y,x) on the other side.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "for your first problem, you have:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = 4^x is the original equation.<BR>\n" );
document.write( "y = log4(x) is the inverse equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the graph of that about the line y = x looks like this.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/100101.jpg\" alt=\"$$$\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "for your second problem, you have:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you get y = 4 * 2.8^x is your original equation.<BR>\n" );
document.write( "y = log(x/4) / log(2.8) is your inverse equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the graph of that about the line y = x looks like this.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/100102.jpg\" alt=\"$$$\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "for your third problem, you have:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you get y = log(x/19) is your original equation.<BR>\n" );
document.write( "y = 19 * 10^x is your inverse equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the graph of that about the line y = x looks like this.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/100103.jpg\" alt=\"$$$\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here's a reference you might find useful.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.purplemath.com/modules/logs2.htm\" target = \"_blank\">https://www.purplemath.com/modules/logs2.htm</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );