document.write( "Question 95480: Please help me find the inverse of the function.
\n" ); document.write( "f(x)= (-5+5x)/(3-x) \n" ); document.write( "
Algebra.Com's Answer #69478 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "Please help me find the inverse of the function.\r\n" );
document.write( "\"f%28x%29=+%28-5%2B5x%29%2F%283-x%29\"\r\n" );
document.write( "\r\n" );
document.write( "The graph of the inverse of a function is the reflection\r\n" );
document.write( "of the graph across the line whose equation is y = x which\r\n" );
document.write( "is called the \"identity line\", because y and x are \r\n" );
document.write( "identically equal on that line.\r\n" );
document.write( "\r\n" );
document.write( "So we'll be able to check that when we finish, we first\r\n" );
document.write( "graph the equation by getting points, plotting them and\r\n" );
document.write( "drawing a smooth curve through them. \r\n" );
document.write( "\r\n" );
document.write( "\"f%28x%29=+%28-5%2B5x%29%2F%283-x%29\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we follow the rules for finding the inverse:\r\n" );
document.write( "\r\n" );
document.write( "1. Replace f(x) by y\r\n" );
document.write( "2. Interchange x and y\r\n" );
document.write( "3. Solve for y\r\n" );
document.write( "4. Replace y by f-1(x)\r\n" );
document.write( "\r\n" );
document.write( "1. Replace f(x) by y\r\n" );
document.write( "\r\n" );
document.write( "\"f%28x%29=+%28-5%2B5x%29%2F%283-x%29\"\r\n" );
document.write( "\"y+=+%28-5%2B5x%29%2F%283-x%29\"\r\n" );
document.write( "\r\n" );
document.write( "2. Interchange x and y\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%28-5%2B5y%29%2F%283-y%29\"\r\n" );
document.write( "\r\n" );
document.write( "3. Solve for y\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%28-5%2B5y%29%2F%283-y%29\" \r\n" );
document.write( "\r\n" );
document.write( "Multiply both sides by \"%283-y%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"x%283-y%29+=+-5%2B5y\"\r\n" );
document.write( "\r\n" );
document.write( "\"3x-xy+=+-5%2B5y\"\r\n" );
document.write( "\r\n" );
document.write( "Get all terms in y on the right and\r\n" );
document.write( "all others on left:\r\n" );
document.write( "\r\n" );
document.write( "\"3x%2B5=5y%2Bxy\"\r\n" );
document.write( "\r\n" );
document.write( "Factor out y on the right\r\n" );
document.write( "\r\n" );
document.write( "\"3x%2B5=y%285%2Bx%29\"\r\n" );
document.write( "\r\n" );
document.write( "Divide both sides by \"%285%2Bx%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"%283x%2B5%29%2F%285%2Bx%29+=+y\"\r\n" );
document.write( "\r\n" );
document.write( "Swap sides for convenience \r\n" );
document.write( "\r\n" );
document.write( "\"y+=%283x%2B5%29%2F%285%2Bx%29\"\r\n" );
document.write( "\r\n" );
document.write( "4. Replace y by f-1(x)\r\n" );
document.write( "\r\n" );
document.write( "f-1(x) = \"%283x%2B5%29%2F%285%2Bx%29\"\r\n" );
document.write( "\r\n" );
document.write( "Now let's graph this on the same set of\r\n" );
document.write( "axes in green:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we'll draw in the identity line whose equation is y = x. \r\n" );
document.write( "We'll draw it dotted just to see if the green graph is the \r\n" );
document.write( "reflection of the dark red one in that dotted identity line:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Yes it is, so f-1(x) = \"%283x%2B5%29%2F%285%2Bx%29\" must be right. \r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );