document.write( "Question 1195227: Suppose a function $\"f\"$ is such that $\"f%281%2Fx%29-3f%28x%29=x\"$ for every x≠0(x that does not equal 0). Find $\"f%282%29\"$ . \n" ); document.write( "

Algebra.Com's Answer #827648 by ikleyn(52798) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Suppose a function f is such that f(1/x)-3f(x)=x for every x≠0 (x that does not equal 0). Find f(2).
\n" ); document.write( "~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Write the given functional equation for x = 2 and for x = 1/2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "For x = 2, you will have\r\n" );
document.write( "\r\n" );
document.write( "    f(1/2) - 3f(2) = 2              (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "For x = 1/2, you will have\r\n" );
document.write( "\r\n" );
document.write( "    f(1/(1/2)) - 3f(1/2) = 1/2,  \r\n" );
document.write( "\r\n" );
document.write( "or\r\n" );
document.write( "\r\n" );
document.write( "    f(2) - 3f(1/2) = 1/2.           (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Let  u = f(1/2),  v = f(2)  for breavity.  Then you can re-write (1) and (2) in the form\r\n" );
document.write( "\r\n" );
document.write( "    u - 3v = 2                       (3)\r\n" );
document.write( "    \r\n" );
document.write( "    v - 3u = 1/2                     (4)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now I want to solve the system of linear equations (3) and (4) and find v.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "For it, I express  u = 2+ 3v  from (3) and substitute it into equation (4)\r\n" );
document.write( "\r\n" );
document.write( "    v - 3*(2+3v) = 1/2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In the last equation, multiply both sides by 2 to run from denominator\r\n" );
document.write( "\r\n" );
document.write( "    2v - 6*(2+3v) = 1\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Simplify and find v\r\n" );
document.write( "\r\n" );
document.write( "    2v - 12 - 18v = 1\r\n" );
document.write( "\r\n" );
document.write( "        -16v      = 1 + 12\r\n" );
document.write( "\r\n" );
document.write( "        -16v      =   13\r\n" );
document.write( "\r\n" );
document.write( "           v      =   .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus  v = f(2) = ,  and the problem is just solved.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  f(2) = .\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved and explained.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );