document.write( "Question 1209327: Let
\n" ); document.write( "f(x) = (2x + 5)/(x - 4).
\n" ); document.write( "If f^{-1} is the inverse of f, what is f^{-1}(1)? \n" ); document.write( "

Algebra.Com's Answer #848369 by ikleyn(52798) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Let f(x) = (2x + 5)/(x - 4).
\n" ); document.write( "If f^{-1} is the inverse of f, what is f^{-1}(1)?
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "All you need to do to find f^{-1}(1)  is to solve this equation for x\r\n" );
document.write( "\r\n" );
document.write( "     = 1.    <<<---===  the sign is corrected after the notice from @greenestamp.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It can be solved in a few lines\r\n" );
document.write( "\r\n" );
document.write( "    2x + 5 = x - 4.\r\n" );
document.write( "\r\n" );
document.write( "    2x - x = -4 - 5\r\n" );
document.write( "\r\n" );
document.write( "       x   =    -9.     \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWEER.  f^{-1}(1) is -9.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "--------------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "What I am trying to explain is that in this problem you do not need
\n" ); document.write( "to restore the inverse function f^{-1}(x) explicitly as a function of x,
\n" ); document.write( "as the other tutor does.   It is unnecessary work.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "All you need to do to find f^{-1}(1) is to solve this equation, f(x) = 1, for x.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In problems of this kind, it is necessary to restore f^(-1)(x) explicitly in two cases: \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "            (1)   if the problem explicitly asks you about it, \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "    and/or\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "            (2)   if the problem asks to calculate f^(-1)(x) for several/many values of x.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Then restoring the inverse function is justified from the point of view of effectiveness of your efforts.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Otherwise, if you are asked to find f^(-1)(x) for one single value of x = c,
\n" ); document.write( "it is more effective to solve an equation f(x) = c for this single value of \"c\".\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );