document.write( "Question 1164405: The functions f and g are defined as follows:
\n" ); document.write( "f(x) = sqrt(x+1/x-1) and g(x) = sqrt(x+1)/sqrt(x-1).
\n" ); document.write( "Explain why functions f and g are not the same function. \n" ); document.write( "
Algebra.Com's Answer #788803 by ikleyn(52785)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "These two functions,  f(x)  and  g(x),  have different domain;  \r\n" );
document.write( "\r\n" );
document.write( "in other words, they are defined in different sets of real numbers.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The domain for function g(x)  is  { x |  x > 1 }.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The domain for function f(x)  is  the set of solutions to THIS inequality\r\n" );
document.write( "\r\n" );
document.write( "    \"%28x%2B1%29%2F%28x-1%29\" > 0,   \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "which is the UNION of two sets\r\n" );
document.write( "\r\n" );
document.write( "    { x < -1 } U { x > 1 },\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "where the numerator and denominator BOTH are negative OR BOTH are positive.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "See this visual illustration, where I slightly changed function g(x) intently to make the difference visible.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Plot  f(x) = \"sqrt%28%28x%2B1%29%2F%28x-1%29%29\" (red), g(x) = \"sqrt%28x%2B1%29%2Fsqrt%28x-1%29\" (green)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    f(x)  has two branches, as you see (two red lines);  \r\n" );
document.write( "\r\n" );
document.write( "    g(x)  has only one branch (only one green line).\r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Answered, solved and explained.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Is everything clear to you in my solution ?\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "`````````````````````````````````\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "If after reading my post you will have questions,  do not hesitate to post them to me.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "If you do,  then please refer to the  ID  number of this problem,  which is  1164405.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The  ID  number is the first number,  which you see in your page in the  UPPER  LEFT  corner.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "If you will post to me without referring to this  ID  number,  I will not know to whom to answer.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "It is how this forum works.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So,  mentioning this ID is  NECESSARY,  if you want to have two-way communication with me.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );